Irreversible monetary policy at the zero lower
bound
Kohei Hasui
Teruyoshi Kobayashi
Tomohiro Sugo
April 2019
Discussion Paper No.1906

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANIrreversible monetary policy at the zero lower bound∗
Kohei Hasui†

Teruyoshi Kobayashi‡

Tomohiro Sugo§

April 15, 2019
Abstract
Real-world central banks have a strong aversion to policy reversals. Nevertheless, theoretical models of monetary policy within the dynamic general equilibrium framework normally ignore the irreversibility of interest rate control. In this paper, we develop a formal
model that incorporates a central bank’s discretionary optimization problem with an aversion to policy reversals. We show that, even under a discretionary regime, the optimal
timing of liftoff from the zero lower bound is characterized by its history dependence, whicharises from the option value to waiting, and there exists an optimal degree of reversal aversion at which the social loss is minimized.
JEL Classification: E31, E52, E58, E61
Keywords: Monetary policy, policy irreversibility, reversal aversion, liquidity trap∗

We thank Yasuo Hirose, Munechika Katayama, Ryo Kato, Taisuke Nakata, Shin-ichi Nishiyama, Kengo
Nutahara, Mototsugu Shintani, Yuichiro Waki, and Tomoaki Yamada for their useful suggestions and comments.
Hasui acknowledges financial support from JSPS KAKENHI Grant Number 17K13768. Kobayashi acknowledges
financial support from JSPS KAKENHI Grant nos. 15H05729 and 16K03551. The views expressed here are thoseof the authors and do not necessarily reflect the official views of the Bank of Japan.
†
Matuyama University. E-mail: khasui@g.matsuyama-u.ac.jp‡
Kobe University. E-mail: kobayashi@econ.kobe-u.ac.jp§
Bank of Japan. E-mail: tomohiro.sugou@boj.or.jp

11

Introduction

Many studies argue that central banks prefer “gradualism” in which the targets for the official
interest rates are smoothed. The hypothesis of gradualism in monetary policy, or “interest
rate smoothing,” has been supported by a number of empirical studies (Sack and Wieland,
2000; English et al., 2003). On the other hand, there are also studies that do not support theinterest-rate-smoothing hypothesis. They show that the statistical significance of policy inertia
may come from an inappropriate estimation procedure, such as omitting serially correlated
variables (Rudebusch, 2002, 2006), ignoring the weak instrument problem (Consolo and Favero,2009) or missing time-varying equilibrium interest rates (Trehan and Wu, 2007).
An important yet often ignored aspect of the interest rate policy in practice is that thedecisions of central banks are rarely reversed within a short period of time (Blinder, 2006;
Mendes et al., 2017). For instance, if the target interest rate is raised by 0.25% today, it isquite unlikely that the central bank would decide to cut it by 0.25% at the next policy meeting.
Rather, the only option left for the central bank would be either to increase the interest rateor to keep it unchanged. Thus, monetary policy decisions are practically irreversible, or moregenerally, central banks have an aversion to policy reversals (Lowe and Ellis, 1997; Blinder, 2006;
Mendes et al., 2017). In the US, the Fed changed the Federal funds target 95 times between
1990 and 2008, and there were only two cases in which the Fed reversed their policy directions
within two quarters. Nevertheless, the discussion of interest rate smoothing has normally been
based on the standard linear-quadratic optimization problem of a central bank or the empiricalresults obtained within a class of the partial adjustment models, in both of which immediate
interest-rate reversals are allowed at no cost. Alan Blinder, former vice chairman of the Boardof Governors of the Federal Reserve System, asserts:“Although the basic logic of optimization suggests that such policy reversals should
not be uncommon, central bankers seem to avoid them like the plague.” (Blinder,
2006)
Blinder (2006) points out three factors that would lead central banks to have reversal aversion; central bankers tend to i) be concerned about losing their credibility, ii) avoid creating
unnecessary volatility in the financial market and iii) be unwilling to be seen as admitting errors.
Given that real-world monetary policies are rarely reversed within a short period of time, a2feasible optimal policy would have to be described as a solution to an optimization problem with
an irreversibility constraint or an aversion to policy reversals. In this paper, we show that thecentral bank’s reversal aversion becomes a source of policy inertia even under a discretionary
regime. The intuition behind the emergence of policy inertia is that there arises an option value
to waiting, in the same way that is discussed in the classical studies on the irreversibility ofinvestment (Bernanke, 1983; Dixit and Pindyck, 1994). Once an investment has been made, thefirm holds a risk of not being able to resell the installed capital at the desired price. The firm
thus may find it optimal to hold off on making an investment and wait to see what will happen
to the capital price in the future. In monetary policy, a central bank needs to absorb shocks to
the natural rate by controlling the interest rate (Woodford, 2003; Walsh, 2017). In the presenceof an irreversibility constraint, however, it may be optimal not to immediately react to thecurrent shocks, because a current contractionary (expansionary) policy will prohibit the central
bank from implementing an expansionary (contractionary) policy in the future. If there is achance that the natural rate would revert to the previous level in the near future, then the valueof option to wait can be larger than the cost of not fully absorbing the current shock. In thisway, the central bank needs to manage the risk of the current policy change itself constraining
future policy decisions.
Within this realistic optimization framework, we ask one of the most important questions inrecent monetary policy: how should a central bank determine the timing of liftoff from the zero
lower bound (ZLB)? (Evans et al., 2015; Carlstrom et al., 2015; Nakata and Schmidt, 2018).
Under discretionary policy, a central bank always has to absorb the natural rate shock, so theinterest rate will be lifted from zero as soon as the natural rate takes a positive value. However,this is suboptimal once the interest rate hits the ZLB, because the zero-interest-rate policy(ZIRP) would be contractionary as long as the natural rate takes a negative value at which
there is a gap between the actual and the desirable interest rates. The fully optimal policy
is to keep the interest rate at zero for an extended period of time even after the natural rate
turns positive (Eggertsson and Woodford, 2003; Jung et al., 2005). However, the well-known
difficulty is that a commitment to the ZIRP is time inconsistent and therefore it is not practical
for real-world central banks to implement the optimal commitment policy (Nakata, 2015).
In this study, we find that the policy inertia stemming from the central bank’s reversal
aversion functions as a commitment device at the ZLB. Since there is a value to waiting, the3central bank would keep the interest rate at zero even after the natural rate turns positive. The
extended duration of the ZIRP thus mimics the optimal commitment policy, which would leadto a lower welfare loss. In fact, we reveal that there exists an optimal degree of reversal aversion
at which the social loss will be minimized. This finding suggests that, in an economy with the
ZLB, it would be desirable for the society to delegate monetary policy to an independent central
banker having an appropriate degree of policy reversal aversion. Our model thus provides analternative and practically feasible delegation scheme for monetary policy (Rogoff, 1985; Walsh,
2003; Bilbiie, 2014). In particular, in contrast to the previous delegation schemes in which a
central bank is assigned additional target variables other than inflation and output, the proposeddelegation scheme based on policy irreversibility is target-free in the sense that no additionaltarget variables are required.
Aside from the theoretical contribution to the literature on interest rate policy at the ZLB,
our study also contributes to the general DSGE framework by proposing a novel technique forsolving a dynamic optimization problem with an irreversibility constraint, or more generally withan aversion to policy reversals. While Kobayashi (2010) solved an optimization problem witha strict irreversibility constraint using the Bellman formulation, the current method employs amore general approach that allows us to treat a strict irreversible control problem as a specialcase. Here, we formulate the central bank’s problem as a dynamic loss-minimization problem
with a penalty for immediate reversals, in which a strictly irreversible policy corresponds tothe solution under a prohibitively high penalty. The constrained monetary policy is analyzed
in an otherwise standard new Keynesian model, so there could be various possible applicationsof this method for the study of recent monetary policy tools such as quantitative easing (Chenet al., 2012; Gertler and Karadi, 2013).

2

Related literature

The current study analyzes the effect of a central bank’s reversal aversion within the standard
new Keynesian framework developed by Woodford (2003). Along with many other studies, ourpaper focuses on discretionary policy because the optimal commitment policy is in principle
time-inconsistent and notoriously difficult to implement in practice (Nakata, 2015). Many
researchers have thus proposed “commitment devices,” which would lead a discretionary policy
to be closer to, or under certain circumstances even identical to, the optimal commitment policy.
4This is also known as the problem of optimal delegation of monetary policy and has been an
active research area over the past three decades (Rogoff, 1985; Walsh, 2003, 2017; Bilbiie, 2014).
For a recent example, Bilbiie (2014) proposes a specific loss function for the central bank withwhich the discretionary policy becomes identical to the optimal commitment policy. Our findingthat there is an optimal degree of reversal aversion suggests that delegating monetary policy
to an independent central banker having an aversion to policy reversals would improve social
welfare. Our study thus contributes to the literature on the optimal delegation problem byproposing a highly feasible delegation scheme.
As mentioned in Introduction, a disadvantage of discretionary policy at the ZLB is thatthe interest rate is controlled to move in sync with the natural rate, if possible, although the
optimal commitment policy requires the interest rate to be kept at the ZLB for an extended
period of time even after the natural rate turns positive (Eggertsson and Woodford, 2003;
Jung et al., 2005; Adam and Billi, 2006, 2007; Nakov, 2008). Accordingly, as a solution to thedelegation problem, various commitment devices have been proposed to allow central banks
to delay the timing of liftoff from the ZLB. As argued by Woodford (2003), the point is how
to effectively incorporate history dependence, or policy inertia, into practical monetary policy.
Nakov (2008) argues that social loss will be reduced by employing a Taylor rule that depends on
the lagged interest rate. Nakata and Schmidt (2018) analyze optimal discretionary policy in aliquidity trap, showing that a central bank with an interest-rate-smoothing objective can lower
social loss by mimicking the optimal commitment policy. Jeanne and Svensson (2007) assertthat a central bank’s concern about its balance sheet and the level of capital would provide amechanism to commit to a low interest rate. Boneva et al. (2018) consider a “threshold-based
forward guidance” in which a central bank makes a state-contingent commitment to the ZIRP
until the values of particular macroeconomic variables exceed prespecified thresholds.
The current paper investigates the role of a central bank’s intrinsic aversion to policy reversals as a commitment device at the ZLB. Although the prevalence of policy reversal aversion
and the need to study it have been repeatedly pointed out by several central bankers (Loweand Ellis, 1997; Blinder, 2006; Mendes et al., 2017), formal analysis of this phenomenon is still
scarce, with the exception of the work by Kobayashi (2010). While Kobayashi (2010) analyzes
strict policy irreversibility using a backward-looking model developed by Ball (1999) and Svensson (1997), the current study employs a more general approach within the forward-looking new5Keynesian model.
An important property that follows from a central bank’s reversal aversion is that there
exists an option value to waiting. A similar mechanism has been well studied in the literatureon the irreversibility of investment (Bernanke, 1983; Dixit and Pindyck, 1994), in which a firm
has to determine the optimal amount of investment, assuming that the installed capital could
not be resold (i.e., “reversed”) at the desired price. Recently, Lei and Tseng (2017) considereda “wait-and-see” monetary policy in which a central bank faced with fixed adjustment costs forpolicy shifts values the option to wait until new information arrives in the future. A crucial
difference from our study is that in their model, the emergence of the value of waiting resultsfrom the exogenous fixed costs of policy changes, while a central bank’s intrinsic reversal aversion
is the only factor that generates the value of waiting in our model. Our model thus makes it
possible to quantify the welfare effect of a central bank’s intrinsic aversion to policy reversals.

3

Policy reversal aversion in practice

As explained in Introduction, central banks tend to avoid reversing their policy decisions once
their new policy actions have been recognized publicly. In the literature, on the other hand, theobserved gradual policy shifts have been interpreted as reflecting the central banks’ objective ofinterest rate smoothing (Sack and Wieland, 2000; English et al., 2003). In this section, we firstdescribe the differences and similarities between the well-examined interest rate smoothing andpolicy reversal aversion. Then, we discuss some empirical evidence that the Fed and the Bankof England really have an aversion to reversals.

3.1

Relationship between interest rate smoothing and reversal aversion

The nominal interest rate under interest rate smoothing is generally specified as a policy functionof the form
it = f (it−1 , Zt , et ) ,(1)

where it is the nominal interest rate (or the policy rate), and Zt and et denote the vectors ofendogenous variables, typically inflation and output, and exogenous variables such as economic
shocks, respectively. Eq. (1) can be regarded as a general functional form for interest ratesmoothing in which the central bank tries to suppress the fluctuation of the change in the level6

of the interest rate, ∆it ≡ it − it−1 . The central bank takes it−1 into account in setting it , but
the direction of the current policy change, given by sgn(∆it ), is determined independently ofsgn(∆it−1 ). For instance, in a widely used partial adjustment model, the current interest rate isexpressed as a weighted average of the past interest rates and the desired value (English et al.,
2003; Rudebusch, 2006).
On the other hand, the policy reversal aversion we will consider in this paper constrains
the direction of policy changes. Specifically, a policy function of a central bank having policy
reversal aversion takes the form:
it = g (it−1 , ∆it−1 , Zt , et ) .

(2)

The central bank now needs to take into account the previous policy shift ∆it−1 , as well as theprevious level of the interest rate it−1 , to avoid reversing the previous direction of the policychange in setting it (or equivalently, ∆it ). In particular, policy irreversibility is regarded as a
special case in which policy reversals are strictly prohibited and thereby sgn(∆it ) · sgn(∆it−1 ) ≥0.
Kobayashi (2010) argues that due to the functional similarity between Eqs. (1) and (2), one
could not statistically identify the difference between the two regimes, interest rate smoothing
and policy irreversibility, if the regression equation for it is based on a partial adjustment model.
If the central bank really has an aversion to policy reversals, however, the sign of ∆it−1 would
affect the sign of ∆it , and thereby the level of the current interest rate it depends on it−1 in anonlinear manner.

3.2

Example: The Fed and the Bank of England

Now let us look at the empirical data on the policy changes of central banks to see if there isa tendency toward reversal aversion. We consider the Federal Funds (FF) target rate and the
Bank rate as the policy instruments of the Fed and the Bank of England, respectively. The dataon the FF target rate ranges from August 1987, when the chairman Greenspan was appointed,
to December 2008. For the UK, the data ranges between June 1997, when the BoE was given
its independence, and December 2016.
Let the sign of a policy change be δ ≡ sgn(∆i). If a central bank has an aversion topolicy reversals, then the distribution P (δ) would be different from the conditional distribution
P (δ|δ− ), where δ− denotes the sign of the last policy shift. By contrast, if the central bank’s
7(a) US
Probability

FF target (%)

10

8

6

4

2

0
1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008

year

P( )

P( |

=

1) P ( |

= 1)

P( )

P( |

=

1) P ( |= 1)(b) UK
7

Probability

Bank rate (%)

8

6
5
4
3
2
1
0

1998

2000

2002

2004

2006

2008

year

2010

2012

2014

2016

Figure 1: Policy rates in the US and the UK. (a) The Federal Funds target rate (left panel)and the distribution P of policy-shift signs δ ≡ sgn(∆i) (right panel). δ− denotes the sign ofthe last policy change. (b) The Bank rate (left panel) and the distributions of the sign of the
BoE’s policy shifts (right panel).
decision is independent of the direction of the last policy change, as in the case of interest ratesmoothing, then it would be that P (δ|δ− ) = P (δ). For simplicity, in this section we only consider
positive and negative policy shifts (i.e., δ ∈ {−1, 1}), since it is hard to identify the timing of“no policy change” (i.e., δ = 0). Because there was always a possibility of having a non-regularpolicy meeting especially in the 80’s, looking only at the regularly held meetings does not fully
capture the decision of the central banks to do nothing (Thornton, 2006; Kobayashi, 2009). It
should be noted that the number of reversal policy changes would be lower if we included theevents with δ = 0. We will consider the possibility of δ = 0 in the theoretical model.
Fig. 1 shows that for both the Fed and the BoE, the unconditional distribution P (δ) is
significantly different from P (δ|δ− ). The observed common patterns regarding the relationshipbetween P (δ) and P (δ|δ− ) are summarized in Table 1. These patterns suggest that the current
direction of policy change is largely affected by the sign of the last policy shift, and there are few

8Table 1: Observed relationship between P (δ) and P (δ|δ− ). See the caption of Fig. 1 for thedefinitions of P , δ and δ− .
δ = −1δ=1

δ− = −1

P (−1) < P (−1| − 1),

P (1) > P (1| − 1)

δ− = 1P (−1) > P (−1|1),

P (1) < P (1|1)

cases in which policy directions are reversed within a short period of time. Note that P (1| − 1)
and P (−1|1) are not zero because we exclude the possibility of δ = 0 as explained above. Thus,
we necessarily observe some policy reversals, but these changes were mostly made after a long
period of “doing nothing.” This property is not captured by the conventional model of interest
rate smoothing, because it allows the policy rate to move in any direction independently of thepast policy changes. In the following sections, we provide a formal framework for analyzing
optimal monetary policy when the central bank has an aversion to policy reversals.

4

Simplified framework: A finite-period model

To see how the introduction of reversal aversion would affect the optimal conduct of monetary
policy, we first consider a simple four-period model with the ZLB. In this section, we assume forsimplicity that the central bank is not allowed to reverse the direction of a policy shift made in
the previous period. We will relax this assumption and consider an arbitrary degree of reversal
aversion in Section 5.

4.1

Structural equations

We consider the following standard new Keynesian model developed by Woodford (2003):
xt = Et xt+1 − σ(it − Et πt+1 − rtn ),(3)πt = βEt πt+1 + κxt ,(4)

for t = 0, 1, 2, 3, where xt , πt , and it denote the output gap, inflation rate, and nominal interest
rate, respectively. Et is the expectations operator conditional on information available at time t.
Equation (3) is the forward looking IS curve, which is derived from the standard Euler equation

9

for consumption (Woodford, 2003; Walsh, 2017). The IS equation indicates that the current
output gap is determined by the expected output gap and the deviation of the real interest
rate from the natural rate, denoted by rtn . Equation (4) is the standard new Keynesian Phillips
curve; the current inflation rate is expressed as the sum of the current output gap and inflation
expectations when firms’ opportunity to change prices is given by a constant probability (Calvo,1983).

4.2

Monetary policy with an irreversibility constraint

Suppose that a central bank faces a strict irreversibility constraint in which reversing the direction of the policy shift made in period t − 1 is not allowed in period t (Kobayashi, 2010). In thepresence of a strict irreversibility constraint, the control space Ωt for the current policy rate it
is given by{it | it ≤ it−1 , it ∈ Ω}Ωt = Ω

{it | it ≥ it−1 , it ∈ Ω}if δt−1 = −1,
if δt−1 = 0,(5)

if δt−1 = 1,

where δt denotes the sign of the current policy shift ∆it ≡ it − it−1 , defined as δt ≡ sgn(∆it ).
Ω denotes the set of all possible values that the policy rate can take. To keep the analysis
simple, we consider a small set of discrete values as the control space: Ω ≡ [0, 0.5, 1, 1.5] × 1/4.
It should be noted that the set Ω does not include negative values, which effectively introduces
the ZLB constraint: it ≥ 0, t = 0, 1, 2, 3 (Eggertsson and Woodford, 2003; Jung et al., 2005;
Adam and Billi, 2006, 2007; Billi, 2011; Coibion et al., 2012; Nakata and Schmidt, 2018).
The social loss function to be minimized by the central bank is given by the sum of thePsquared deviations of inflation and of the output gap from their steady states: 12 E0 3t=0 β t πt2 + λx2t .
Thus, the central bank’s minimization problem constrained by the irreversibility condition isformulated as3


1 X t 2E0
β πt + λx2t ,it ∈Ωt 2
min

(6)

t=0

subject to Eqs. (3) and (4). Unlike the standard models of monetary policy, the central bankis faced with a time-varying control space, which suggests that the choice of the current policy
rate is not independent of the change and the level of the previous policy rate. This adds a non10

trivial nonlinearity to the model in addition to the ZLB constraint. We solve the minimization
problem by formulating the value function.

4.3

Solving the model

We solve the four-period model in a backward manner. The central bank’s problem in the lastperiod (i.e., t = 3) can be specified by the value function:
V3 (r3n , i2 , δ2 ) = min π32 + λx23 ,i3 ∈Ω3s.t. x3 = −σ(i3 − r3n ),π3 = κx3 ,

(7)
(8)
(9)

where Vt denotes the value at time t. Because the economy terminates at t = 3, the expectations
terms in the IS and the Phillips curves are dropped (Nakata et al., 2018). For t = 0, 1, 2, on
the other hand, we need to include the expectations terms obtained in the previous step. The
central bank’s problem thus leads tonVt (rtn , it−1 , δt−1 ) = min πt2 + λx2t + βEt Vt+1 (rt+1, it , δt ),it ∈Ωts.t. xt = Et xt+1 − σ(it − Et πt+1 − rtn ),πt = βEt πt+1 + κxt ,

(10)
(11)
(12)

for t = 0, 1, 2. We assume i−1 = δ−1 = 0 to consider a situation in which the economy is alreadyin a liquidity trap at the beginning of t = 0.
4.3.1

Parameters

Consider a simple scenario in which the natural rate takes either of the following two values:rL or rH , where rL < rH . The behavior of the natural rate is ruled by a Markov process whose
conditional probabilities are given bynProb(rtn = rL |rt−1= rL ) = q,(13)nProb(rtn = rL |rt−1= rH ) = p.

(14)

In this section, we set rL = −1/4 and rH = 1.5/4. Note that to investigate the optimal policy
decision at the ZLB, rL should be sufficiently small so that the policy rate could hit the ZLB.
11

Table 2: Realized path of the natural rate in the four-period model
t

0

1

2

3

rtn

rL

rH

rH

rH

Following Adam and Billi (2007), the other model parameters are specified as follows: β = 0.99,
σ = 1, κ = 0.024, λ = 0.003 and q = 0.875. To see the uncertainty effect stemming from thenatural rate, we will examine different values of p in the following.
To obtain an intuition behind the optimal (discretionary) irreversible policy at the ZLB, we
specify the realized path of the natural rate as in Table 2. This scenario allows us to examine
an illustrative situation in which the central bank is trapped at the ZLB at t = 0 but notnecessarily for the periods from t = 1 onward.
4.3.2

Irreversible policy as risk management

As mentioned above, we consider that the economy has already been in a liquidity trap at the
beginning of t = 0 so that the irreversibility constraint does not bind at t = 0. Since a positive
natural rate shock hits the economy in period 1 (i.e., r1n = rH ), the problem for the central
bank at t = 1 is whether to raise the interest rate (i.e., i1 > 0) or to keep it at the ZLB (i.e.,
i1 = 0). Let `t ≡ πt2 + λx2t be the temporal loss for the central bank. In period 1, the net benefitof raising the interest rate (i.e., i1 > i0 = 0) relative to keeping i1 at 0 is given by
N1 = `1 |i1 =0 − `1 |i1 >0 + βE1 [V2 (r2n , 0, 0) − V2 (r2n , i1 , 1)]= `1 |i1 =0 − `1 |i1 >0 + β {p [V2 (rL , 0, 0) − V2 (rL , i1 , 1)]+ (1 − p) [V2 (rH , 0, 0) − V2 (rH , i1 , 1)]} .

(15)

The central bank raises the interest rate in period 1 if N1 > 0, or`1 |i1 =0 − `1 |i1 >0|{z}

Temporal benefit of raising i1> β E1 [V2 (r2n , i1 , 1) − V2 (r2n , 0, 0)] .
|
{z
}

(16)

Expected cost of raising i1

The LHS of Eq. (16) represents the temporal benefit of lifting off from the ZLB, which takes
a positive value since the central bank could absorb the current natural rate shock by raising
i1 , leading to a lower temporal loss. The RHS of Eq. (16) is interpreted as the (discounted)
expected cost of raising i1 . Increasing the current interest rate might cause losses in the future
12because there is a possibility that a negative shock would occur again (i.e., r2n = rL ). In thatcase, the central bank would need to lower the interest rate in period 2 to absorb the negative
shock, reversing the direction of the policy change decided at t = 1. However, such a reversal
policy shift is not allowed due to the irreversibility constraint. To maintain the controllabilityof the interest rate in period 2, the current interest rate needs to be kept at the ZLB in period
1 (i.e., i1 = δ1 = 0). Thus, the central bank faces a trade-off between the current benefit ofabsorbing the existing shock and the expected cost of abandoning flexible policies in the future.
An alternative way to interpret the optimal policy decision in period 1 is to regard thecost of not raising i1 as the price of an option, in analogy with the irreversibility of investment (Bernanke, 1983; Dixit and Pindyck, 1994). If the central bank raises the interest rate
in period 1, it would lose the option to cut the interest rate in period 2, due to irreversibility,
which would cause losses if a negative shock hits the economy in period 2. The temporal loss
that the central bank has to incur when keeping i1 at the ZLB is therefore valuable as it gives
the central bank an option to set i2 = 0 when r2n = rL . Thus, the choice of whether to raise i1
can be considered as a problem of whether to purchase the option to cover the risk of failing toabsorb a negative shock.
The condition that the central bank purchases the “option” (i.e., setting i1 = 0) leads to`1 |i1 =0 − `1 |i1 >0 < β E1 [V2 (r2n , i1 , 1) − V2 (r2n , 0, 0)] .
|
{z
}
|
{z
}
option price(17)

Expected payoff from option

The LHS of Eq. (17) can be interpreted as the price of the option that guarantees a right toconduct flexible monetary policy in period 2 whatever happens to the natural rate. On the other
hand, the RHS corresponds to the expected payoff from the option. The expected payoff fromkeeping i1 at the ZLB, or the option value, plays a role as a price threshold below which thecentral bank is willing to buy the option to cover the irreversibility risk. This risk-management
aspect of the central bank’s problem adds one more dimension to the conventional formulationof monetary policy in which it is implicitly assumed that the central bank can freely control
interest rates. In our model, the central bank needs to manage the risk of the current policychange itself constraining future policy decisions.

13

1.5

1

0.5

0
0

0.2

0.4

0.6

0.8

1

Figure 2: Optimal interest rate in period 1 under irreversible policy vs. p = Prob(r2n = rL |r1n =rH ). Vertical dotted line denotes the threshold of p below which N1 > 0 (See, Eq. (15)).
4.3.3

Optimal irreversible monetary policy

Because a change in policy itself creates a risk, the optimal level of the interest rate depends
on the probability that the natural rate will return to the current value. If the central bankknows in period 1 that the natural rate will return to rL in period 2 with high probability,
then the central bank would not take a risk of losing the option to lower the interest rate
in period 2. Thus, there is a negative relationship between the desired interest rate i1 andProb(r2n = rL |r1n = rH ) = p (Fig. 2). The figure indicates that the central bank should keep i1
at zero if p ≥ 0.4.
Now let us look at the path of the interest rates. For comparison, we also consider thefollowing alternative policy regimes: i) optimal commitment, ii) pure discretionary policy andiii) interest rate smoothing. Each of these three policies is assumed to be conducted without
an irreversibility constraint. To obtain the path under interest rate smoothing, we consider thefollowing standard temporal loss function (Nakata and Schmidt, 2018; Debortoli et al., 2018):222`∆
t = πt + λxt + λ∆ ∆it ,(18)

where λ∆ is set at 0.01 for the moment. To illustrate the uncertainty effect that the probability
p would have on the optimal policy rate, we consider two cases: p = 0.01 (low uncertainty)
14

(a)

(b)

Figure 3: Nominal and real interest rates under different policy regimes. (a) rH is nearly annabsorbing state: p = Prob(rtn = rL |rt−1= rH ) = 0.01. (b) There is a reasonable probability of

the natural rate returning to rL from rH : p = 0.4.
and p = 0.4 (high uncertainty). We note that setting p = 0.01 would be virtually equivalent toassuming that rH is an “absorbing state” in which it is highly unlikely that the natural rate will
return to rL (Eggertsson and Woodford, 2003; Christiano et al., 2011).
As shown in Fig. 3, imposing a higher value of p makes the irreversible policy more history
dependent, as is already indicated in Fig. 2. In other words, the risk of reversing the current
policy decision in the future deters the central bank from controlling the interest rate in a flexible
manner, delaying the timing of liftoff from the ZLB compared to the case of pure discretion(Fig. 3b). On the other hand, under interest rate smoothing, the timing of liftoff is the sameas that under pure discretion, while changes in the interest rate are more smoothed.
This exercise illustrates an important property–i.e., that the irreversibility constraint acts15

as a commitment device at the ZLB. Because the irreversible policy is more inertial than thepure discretionary policy, the behavior of the interest rate mimics that of optimal commitment
at least to some extent (see, Fig. S1 in Supplementary Information (SI) for the responses ofinflation and output). The introduction of an irreversibility constraint could thus reduce social
loss, but the strict irreversibility constraint considered in this section may not be a desirable
choice. Rather, a more flexible policy constrained by an aversion to policy reversals would
further reduce social loss if we could tune the degree of reversal aversion. We will address thisissue in the next section.

5

General case: An infinite-horizon model with an arbitrary degree of reversal aversion

Now let us consider a general model in which the time horizon is infinite. In this section, we
introduce a penalty term in the central bank’s objective function to allow for an arbitrary degreeof reversal aversion. This specification incorporates policies under a strict policy irreversibility
constraint and pure discretion as two polar cases.

5.1

Model

5.1.1

Structure of the model

As in Section 4, we borrow the standard new Keynesian framework with the ZLB:
xt = Et xt+1 − σ(it − Et πt+1 − rtn ),(19)

πt = βEt πt+1 + κxt ,(20)n
rtn = ρr rt−1
+ t ,(21)it ≥ 0.

(22)

Here, the natural rate is assumed to follow an AR(1) process whose disturbance t is given
by an i.i.d. random variable with mean zero and variance σ . β, σ, κ and ρr are parameterssatisfying 0 < β < 1, σ > 0, κ > 0 and |ρr | < 1, respectively.
The social loss function is given by
L = Et∞X
j=02
β j (πt+j+ λx xt+j )2 , λx ≥ 0,

16

(23)

This loss function (23) is obtained in a micro-founded general equilibrium model with sticky
prices (Calvo, 1983) within which the forward-looking IS curve (19) and the new KeynesianPhillips curve (20) are also derived (Woodford, 2003; Yun, 2005; Walsh, 2017).
5.1.2

Central bank with policy reversal aversion

To capture an arbitrary level of policy reversal aversion, here we introduce a penalty term in
the periodic loss function:
Lt = πt2 + λx x2t + λir F (δt−1 , ∆it ),(24)

where F (δt−1 , ∆it ) is given by
F (δt−1 , ∆it )=δt−1 (1 − δt−1 )δt−1 (1 + δt−1 )× [min(∆it , 0)]2 −× [max(∆it , 0)]2 .
2
2

(25)

Eq. (25) simply states that the penalty term F can take one of the following three forms
depending on the sign of the policy shift in period t − 1, δt−1 :[max(∆it , 0)]2 if δt−1 = −1,F (δt−1 , ∆it ) = 0
if δt−1 = 0,





[min(∆it , 0)]2 if δt−1 = 1.

(26)

Note that the central bank is not penalized if there was no policy shift in the last period (i.e.,δt−1 = 0). If the central bank cut (raised) the policy rate in the previous period, then it
would be penalized if it raises (cuts) the policy rate in the current period. The penalty term
thus effectively introduces a “soft” irreversibility constraint, allowing us to capture an arbitrary
degree of reversal aversion by tuning the weight parameter λir . Clearly, the solution to the
problem under a strict irreversibility constraint will be recovered if λir is large enough, in whichcase our formulation is equivalent to the well-known penalty function method for solving a
constrained optimization problem (cf., Luenberger and Ye, 2016, Ch. 13). By contrast, setting
λir = 0 recovers the standard new Keynesian model with the ZLB (Eggertsson and Woodford,
2003; Jung et al., 2005; Adam and Billi, 2007; Nakov, 2008). We also consider an absolute-value
penalty term in section 5.4.

17

Table 3: Baseline parameter values
Parameter

Value

Description

β

0.996

Subjective discount factor

i∗

1.5/4

Nominal interest rate at the steady stateσ

1Intertemporal elasticity of substitution for consumption

κ

0.024

Slope of the Phillips curve

ρr

0.6

Persistence of natural rate shocks

σ

0.233

Standard deviation of natural rate shocks

λx

0.003

Weight on output gap

λir

0.0016

Degree of policy-reversal aversion (optimized)

λ∆0.0008Degree of interest rate smoothing (optimized)

We formulate the central bank’s problem as the Bellman equation of the form
V (St ) = min π(St )2 + λx x(St )2 + λir F (δt−1 , ∆it ) + βEt V (St+1 ),it ≥0s.t. x(St ) = Et x(St+1 ) − σ(it − Et π(St+1 ) − rtn ),(27)π(St ) = κx(St ) + βEt π(St+1 ),where St denotes the state of the economy given by St ≡ [it−1 , δt−1 , rtn ]> . In the following, we
solve the problem by using a value function iteration method (cf., Miranda and Fackler, 2004).
Details about the numerical algorithm are provided in Appendix.
5.1.3

Model parameters

The baseline parameter values are listed in Table 3. The parameter for the intertemporalelasticity of substitution for consumption σ is set at 1 following Adam and Billi (2007). The
coefficient on the output gap in the Phillips curve and the weight on the output gap in theloss function are specified as κ = 0.024 and λx = 0.003, respectively, following Rotemberg andWoodford (1998). As for the parameters related to the natural rate of interest, we set ρr = 0.6
and σ = 0.233 so that we can avoid using extrapolation in approximating the value function.
The discount factor is set at β = 0.996 to ensure that the steady-state value of the nominalinterest rate is 1.5% annually. The key parameter of the model, namely the weight on the
18

(b)

(a)

Figure 4: Policy function for interest rate. Black solid line denotes a pure discretionary policy
without reversal aversion (i.e., λir = 0). We set it−1 = i∗ = 1.5/4 (dotted vertical line).
penalty term λir , is set at 0.0016 since this turns out to minimize the social loss as we will show
below.

5.2

Results

5.2.1

Policy function

In the presence of an aversion to policy reversals, the optimal policy rate under discretion isexpressed as a nonlinear function of the previous interest rate it−1 , the sign of the previous
policy shift δt−1 and the current natural rate rtn :it = i(it−1 , δt−1 , rtn ).

(28)

An important point is that, even under discretionary policy with no intrinsic economic inertia, the current policy rate is largely affected by the previous policy decision, leading to theemergence of an endogenous policy inertia. This suggests the possibility that the determination of the interest rate under discretionary policy would exhibit a history dependence, a key
characteristic of optimal commitment policy.
For comparison, let us first consider a situation in which the penalty for policy reversals is
prohibitively high so that the central bank virtually faces an irreversibility constraint as in theprevious section. With our parameter configurations, we find that the central bank will never
reverse the direction of policy shifts during two consecutive periods if λir ≥ 10 (Fig. 4a). It
19

turns out that the introduction of policy irreversibility makes inflation and output more volatilecompared to those under pure discretion (Fig. S2a).
Fig. 4b illustrates a policy function when reversals are appropriately penalized (i.e., λir =0.0016). Similar to the case of a strict irreversibility constraint, the optimal response to a
positive natural rate shock is weaker when there is an interest rate cut in the previous period(i.e., δt−1 = −1) than when the interest rate is kept unchanged or increased (i.e., δt−1 = 0 or
1). Accordingly, the current inflation and output gap take larger values than in the case of purediscretion when δt−1 = −1 (Fig. S2b).
An interesting property of the policy function with reversal aversion is that the interest
rates with δt−1 = 0 and 1 are both lower than that under pure discretion (Fig. 4b). Note that if
δt−1 = 0 or 1, the central bank would not be penalized by increasing the interest rate at t, andit would therefore be able to fully absorb a positive shock to the natural rate. Nevertheless,it can be optimal not to fully respond to a positive shock, because the higher the current
interest rate, the more likely that the interest rate would need to be cut in the future. This
property illustrates the emergence of a “conservative” or precautionary behavior of the central
bank stemming solely from reversal aversion.
5.2.2

Impulse response

Here, we consider the following two different scenarios for impulse response analysis. In thefirst scenario, the natural rate drops significantly in period 1 so that the nominal interest rate
hits the ZLB, and then the natural rate reverts to its steady state value in period 2. We callthis scenario a temporal liquidity trap. In the second scenario, the natural rate drops in period
1 as in the first scenario, and it stays at the low level in period 2 and then reverts to its steadystate in period 3. We call this situation a prolonged liquidity trap. In both scenarios, we assumeδ0 = 0 and that shocks are not persistent (i.e., ρr = 0) to extract the endogenous persistence
coming from the policy regime itself. As before, we also examine an interest-rate-smoothing
policy, but the weight on the smoothing term, denoted by λ∆ , is now determined such that thesocial loss is minimized.
Figure 5 plots impulse responses for the scenario of a temporal liquidity trap. Two points
are worth mentioning. First, although the interest rate is raised in period 2 for all the examined
regimes, the pace of increasing the interest rate is slower under the optimal discretionary policy

20

(a)

(b)

(c)

(d)

Figure 5: Impulse responses to a temporal −3% natural rate shock. We set ρr = 0 to remove
exogenous policy inertia.
with reversal aversion than under pure discretion (Fig. 5c). This is intuitive because raising
the nominal interest rate in period 2 is penalized due to the reversal aversion term in theloss function. Second, the presence of reversal aversion slightly mitigates the sharp drops in the
output gap and inflation that we see under pure discretion at t = 1 (Fig. 5a, b). This is because,with reversal aversion, private agents know that the central bank’s incentive to raise the interest
rate in period 2 will be partially suppressed, which leads them to expect a higher inflation rate
and output gap. We obtain similar results in the second scenario, namely a prolonged liquidity
trap (Fig. S3).
Although policy irreversibility can be socially beneficial when the interest rate is at the ZLB
by creating an endogenous policy inertia, there is also a cost of not being able to promptly
respond to shocks. For instance, it is optimal under the commitment policy to fully absorb a

21

(a)

(b)

(c)

(d)

Figure 6: Impulse responses to a temporal 3% natural rate shock. We set ρr = 0.6.
positive natural rate shock as long as the interest rate is not constrained by the ZLB (Fig. 6).
However, even for positive shocks, an aversion to policy reversals would prohibit the central bankfrom raising the interest rate enough to completely offset the shocks, because doing so increases
the likelihood that the interest rate would need to be cut in the future. This conservative
policy creates an additional volatility of inflation and output and increases the social loss.
Consequently, reversal aversion can be socially beneficial only in an environment in which the
ZLB is relevant and thereby some extent of policy inertia is required. This gives us a trade-off
between the benefit of creating an endogenous policy inertia at the ZLB and the cost of excess
volatility caused by reversal aversion. In fact, we show in section 5.3.1 that there is an optimal
degree of reversal aversion at which the social loss is minimized.

225Interest rate

4

3

2

1

0
0

5

10

15

20

25

30

35

40

Quarters
Figure 7: Simulated path of nominal interest rate.
5.2.3

Simulated path

Figure 7 illustrates sample simulated paths of the interest rates for λir = 0 (pure discretion),
λir = 10 (strict irreversible policy) and λir = 0.0016 (the policy with the optimal degree ofreversal aversion). We note that the bandwidth within which the interest rate fluctuates becomesnarrower as the penalty weight λir increases. It turns out that the average variance of theinterest rate is 0.945 for λir = 0.0016 and 1.271 for λir = 0. This is intuitive because even in a“normal” circumstance in which the central bank can shift the interest rate freely in the desired
direction, it exhibits a more conservative behavior in that the size of the policy shift is smaller
than that under pure discretion. This reflects the fact that the central bank with reversal
aversion internalizes the influence that the current policy shift would have on the chance thatthe direction of a policy shift would need to be reversed in the future.

5.3

Welfare consequence of policy reversal aversion

The current general equilibrium framework allows us to ask to what extent a central bank’s
policy reversals should be penalized. If policy reversals should not be penalized to flexibly react
to economic shocks, then the desired value of λir will be zero. By contrast, if penalizing policy
reversals would increase social welfare, then the optimal value of λir will take a positive value.

23

30

Welfare gain (%)

20

10

0

-10

-20

-30
0

0.002

0.004

0.006

0.008

0.01

Figure 8: Welfare gain from reversal aversion. The welfare gain, defined by Eq. (29), is maximized at λ∗ir = 0.0016.
As discussed above, a central bank with reversal aversion internalizes the possible influence ofa current policy change on the future reversal probability. Searching for an optimal value of λir
is therefore essentially equivalent to finding the optimal extent of internalization or forwardlookingness of the central bank.
We define the relative welfare gain from reversal aversion as discLW≡−
1× 100,
Lir(29)

where Lir and Ldisc denote the unconditional welfare losses under discretion with and without
reversal aversion, respectively.
5.3.1

Optimal degree of reversal aversion

The welfare gain from reversal aversion against λir is presented in Fig. 8. It is important tonote that there is no welfare gain from reversal aversion in an environment in which there is no
ZLB, i.e., the ZLB constraint is ignored (red dotted line in Fig. 8). In our model the naturalrate is the only source of economic fluctuations, which could be fully offset if there were no
ZLB (a phenomenon called “divine coincidence” (Blanchard and Galí, 2007)). Penalizing policy
reversals in the absence of the ZLB therefore just worsens welfare by prohibiting the central
bank from absorbing the natural rate shocks.
24

(a)

(c)

Welfare gain (%)(b)

Number of periods hitting ZLB under pure discretion
Figure 9: Relationship between welfare gain from reversal aversion and the frequency of hitting
ZLB under pure discretion. Each dot represents a simulation of 10,000 periods.
Fig. 8 confirms that the benefit of penalizing policy reversals could arise only when there isa chance that the policy rate would hit the ZLB, in which case the natural rate shock would
affect the real economy under any kind of policy regime. As shown in Fig. 8, an increase in theweight λir has a non-monotonic impact on the relative welfare gain. For a small value of λir ,the benefit of penalizing the central bank exceeds the cost of inhibiting flexible policy making,but the relative balance between the benefit and the cost is reversed when λir is large enough.
It turns out that there is a unique maximizer λ∗ir at which the social loss is minimized. We also
compare the welfare gains under various policy regimes, such as optimal commitment policy,
simple interest rate rules and interest rate smoothing, in section 5.4. In the following sections
we will examine the detailed mechanisms behind the optimality of reversal aversion.
5.3.2

Frequency of ZIRPs

Because the natural rate is the only source of economic fluctuations, in our model the optimalvalue of λir should be 0 if the ZLB were absent. The benefit of assigning a positive weight on
the penalty term thus depends on the extent to which the ZLB constrains monetary policy. In
fact, our numerical simulations suggest that there is a strong positive correlation between thewelfare gain and the frequency of hitting the ZLB under pure discretion for a given value of λir(Fig. 9). This suggests that the more frequently the central bank is likely to be trapped by the
ZLB, the higher the welfare gain from penalizing policy reversals. It should be noted that the

25

1.7

120

1.6

100

1.5

80

1.4

60

1.3

40

1.2

20

1.1

0

Number of periods

Number of periods

140

1
0

2

4

6

8

10
10-3

Figure 10: Interval and duration of zero interest rate policies (ZIRPs). For a given λir , theaverages are taken over 1,000 simulations with length 10,000 periods.
frequency of hitting the ZLB in Fig. 9 is independent of the degree of reversal aversion.
While the benefit of policy reversal aversion is attributed to the distortions stemming fromthe ZLB, the frequency with which the interest rate hits the ZLB would also be affected by
the degree of reversal aversion. This suggests that there is an endogeneity in the relationshipbetween the frequency of hitting the ZLB and the desirable degree of reversal aversion. Basically,
with other things being equal, imposing a higher weight on the penalty term makes the central
bank more conservative, and thereby the chance that the policy rate reaches the ZLB will belowered. This is reflected by a positive relationship between λir and the average interval time
between ZIRPs (see, bars in Fig. 10). The average interval between ZIRPs is given by theaverage number of periods between the end of a ZIRP and the beginning of the following ZIRP.
In contrast, the relationship between λir and the duration time of a ZIRP is not monotonic(see solid line in Fig. 10). The duration of a ZIRP is defined as the number of periods during
which the interest rate is kept at the ZLB. On one hand, a higher value of λir makes the central
bank less aggressive, as mentioned above, which would make it more likely to keep the policy
rate at zero once it reaches the ZLB. This effect would extend the average duration time ofa ZIRP. On the other hand, the fact that the frequency of hitting the ZLB decreases with26

λir means that the policy rate hits the zero floor only when there is an infrequent yet large
negative shock. If this happens, such a large negative shock is expected to quickly revert tothe steady state in the succeeding periods, which would lead the central bank to exit from the
ZLB shortly. Due to these two opposite effects, the average duration time of a ZIRP exhibits
an inverse U-shaped curve.

5.4

Robustness check

As a robustness check, we will now examine how the optimality of reversal aversion could be
affected by changing the baseline setting. We first consider the influences of varying the valuesof the steady-state interest rate, i∗ , and the weight on the output gap, λx . Then, we compare
the welfare gains under alternative policy regimes.
5.4.1

Steady-state interest rate

As discussed above, the desirability of policy reversal aversion depends heavily on the likelihoodthat the ZLB constraint is binding. The most important parameter that governs the frequencyof hitting the ZLB is the steady state interest rate i∗ . If i∗ is high enough, even a very large
negative shock could not cause the nominal interest rate to reach the ZLB. In contrast, a loweri∗ itself makes it more likely for the central bank to implement a ZIRP.
Fig. 11a shows that there is a negative relationship between i∗ and λ∗ir . This is expectedbased on Fig. 9, which plots a positive correlation between the welfare gain and the likelihoodof being trapped at the ZLB, measured by the number of periods hitting the ZLB under purediscretion. As a rise in i∗ reduces the frequency of reaching the ZLB, imposing a penalty on
policy reversals would become less desirable. It turns out that if the steady-state interest rate
is close to 2%, the advantage of reversal aversion would virtually disappear (i.e., λ∗ir ≈ 0).
5.4.2

Weight on output gap

Because the degree of reversal aversion λir is given as a weight in the loss function, its optimalvalue necessarily depends on the weights on the central bank’s target variables– namely, inflationand output. Fig. 11b plots λ∗ir as a function of the weight on output λx . It shows a positive
relationship between λx and λ∗ir ; as the weight on output increases, the desirable degree ofreversal aversion also rises. This is intuitive because λ∗ir is essentially the weight on policy

27

(b)

Optimal degree of reversal aversion,(a)Figure 11: Optimal degree of reversal aversion. λ∗ir as a function of (a) the steady state interest
rate and (b) the weight on output gap.
reversals relative to inflation and output. λ∗ir needs to be positively correlated with λx to keep
an appropriate level of “punishment” for reversals.
5.4.3

Alternative policy regimes

Next, we compare the welfare performance of optimal irreversible policy with those of different
policy regimes, namely optimal commitment policy, the Taylor rule, interest rate smoothing anda simple interest-rate rule with price-level targeting. We investigate price-level targeting basedon the argument put forth by Eggertsson and Woodford (2003) and Fujiwara et al. (2013) that
introducing a price-level target into a policy rule can be effective when the nominal interest rateis constrained by the ZLB.
We specify the Taylor rule and a simple rule with price-level targeting as follows:• Taylor rule:it = max{0, ρi it−1 + (1 − ρi )(φπ πt + φx xt )},where we set φπ = 1.5 and φx = 0.5. For ρi , we examine two values: ρi = 0 and 0.5.

28

(30)

Table 4: Welfare losses under alternative policy regimes
Policy regime

Welfare loss

Optimal commitment

0.0033

Pure discretion

0.0197

Irreversible policy, λir = 0.0016

0.0154

Irreversible policy, λabs
ir = 0.0008

0.0168Interest rate smoothing, λ∆ = 0.0008

0.0122Taylor rule, ρi = 00.1712Taylor rule with smoothing, ρi = 0.5

0.1365

Simple rule with price-level targeting

0.0851

Note: The values of λir , λabs
and λ∆ are optimized.
ir• Simple rule with price-level targeting:it = max{0, φp (pt − p∗ ) + φx xt },(31)

where φp = 1.5 and φx = 0.5. pt and p∗ denote the logarithms of the price level and its
target value, respectively.
Within a discretionary regime, we also examine a different specification of the penalty term
F in the loss function (24). Instead of penalizing the squared value of ∆it , we consider thefollowing absolute-value penalty F abs :|max(∆it , 0)| if δt−1 = −1,



abs
F (δt−1 , ∆it ) = 0
if δt−1 = 0,





|min(∆it , 0)| if δt−1 = 1.

(32)

Then, monetary policy is delegated to a central bank whose loss function is given byabs
Labs= πt2 + λx x2t + λabs
(δt−1 , ∆it ).
t
ir F

(33)

Table 4 summarizes the social losses under the examined regimes. It turns out that a
discretionary policy with the optimal degree of reversal aversion yield a lower loss comparedto the Taylor rules and a simple rule with price-level targeting, but the optimized interest rate29

smoothing outperforms all the examined policies other than the optimal commitment policy.
As we saw above, interest rate smoothing can incorporate policy inertia by making the previous
interest rate a current state variable through the penalization of ∆it 2 . Our result suggests thatminimizing the squared deviation of ∆it independently of sgn(∆it−1 ) may be more efficientthan penalizing policy reversals for the purpose of reducing the social loss.
However, it is important to note that the welfare performance computed here does not take
into account the potential central bank’s loss of reputation, which could be a serious concern
for real-world central banks. As is pointed out by Blinder (2006), in reality central banks tend
to avoid policy reversals “like the plague” based on the notion that immediate policy reversals
would necessarily undermine their credibility. Given this fact, the degree of feasibility would
have to be evaluated in quantifying the desirability of a policy scheme, a task we leave for future
research.

6

Concluding remarks

Avoiding policy reversals is a common practice of central banks, but the consequent economic
outcome and welfare effects have not been examined so far despite the recent development oftheoretical models of monetary policy. Our study contributes to the literature by revealing thatreversal aversion can be desirable for the society as long as the ZLB is relevant. This implies
that a central bank should have an aversion to policy reversals from the welfare point of view,which provides a rationale for the widely observed phenomenon that policy reversals are rare.
There are some unsettled issues that should be addressed in future research. First, although
the actual policy shifts in developed countries appear to be consistent with the hypothesis thatthese central banks face irreversibility constraints, this may also reflect the fact that the naturalrates themselves move in a way that entails few reversal shifts. To identify to what extent thegradual behavior of interest rates can be explained by the reversal aversion of the central banks,
an estimation of the actual natural rate level within a model with policy reversal aversion would
be needed.
Second, because the introduction of an irreversibility constraint would generally make itharder to guarantee the determinacy of equilibrium, a detailed analysis on the determinacy
condition will be needed. An irreversibility condition, or more generally a penalty for policy
reversals, would occasionally constrain the direction of future policy shifts, so a necessary condi30

tion for local stability would become harder to satisfy compared to the standard models. Third,
the presence of reversal aversion may provide a microfoundation for “interest-rate smoothing.”
As mentioned in Introduction, the standard specifications of interest-rate smoothing in empirical and theoretical models allow for policy reversals and are therefore not necessarily consistent
with the widely observed reversal aversion. Instead, the prevailing gradual behavior of interest
rates may be explained, at least to some extent, by the presence of reversal aversion. More detailed analysis is needed on these issues, and we hope our model will stimulate further research.

Appendix
Numerical algorithm
We solve the central bank’s optimization problem Eq. (27) by value function iteration. Since the
ZLB introduces nonlinearity into the model, unlike in a simple linear-quadratic framework, weneed to employ an approximation technique to obtain the functional form of the value function.
We first specify the grids for the three state variables it−1 , δt−1 , and rtn . Let s1 , s2 , and s3
denote the vectors of grids for it−1 , δt−1 , and rtn , respectively, where the size of si is ni × 1. The
whole state space S is given by a tensor product of the three grid vectors: S = s1 ⊗ s2 ⊗ s3 . The
size of tensor S is then given by N = n1 × n2 × n3 = 129 × 3 × 30. The p.d.f. of the naturalrate is assumed to be normal and discretized into 13 values using the Gaussian quadrature.
Let V (St ) and ht ≡ (xt , πt )> denote a real-valued value function and the vector of forward-

looking variables at time t, respectively, where St ≡ [it−1 , δt−1 , rtn ]> ⊂ S. We compute the valuefunction V and a policy function h as time-invariant functions of S ⊂ S. The procedure is asfollows:
1. Given a particular set of grids for the three state variables, Sjt ⊂ S, and the initial guess
for functions V and h, respectively denoted by V 0 and h0 , compute V 1 (Sjt ), h1 (Sjt ) anda policy function i(Sjt ) as a solution to the problem (27). A cubic-spline function is used
to approximate V 0 (Sjt+1 ) and h0 (Sjt+1 ).
2. Repeat step 1 for all j = 1, . . . N .
3. Stop if kV 1 (S) − V 0 (S)k∞ /kV 0 (S)k∞ < 1.5 × 10−6 . Otherwise, update the functions as
V 0 ≡ V 1 and h0 ≡ h1 , and go to step 1.
31

In our calculation, convergence is reached in four hours using Matlab with Xeon 3.60GHz and
a 32GB memory.

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Supplementary Information“Irreversible monetary policy at the zero lower bound”
Kohei Hasui, Teruyoshi Kobayashi, Tomohiro Sugo

(a)

(b)

Figure S1: Output gap and inflation rate under different policy regimes. (a) rH is nearly annabsorbing state: p = Prob(rtn = rL |rt−1= rH ) = 0.01. (b) There is a reasonable probability of

the natural rate returning to rL from rH : p = 0.4.

1

(a)

(b)

Figure S2: Policy functions for inflation and output gap. Black solid line denotes a purediscretionary policy without reversal aversion (i.e., λir = 0). We set it−1 = i∗ = 1.5/4 (verticaldotted line).

2

(a)

(b)

(c)

(d)

Figure S3: Impulse responses to a persistent −3% natural rate shock in the infinite-horizon
model. We set ρr = 0 to remove exogenous policy inertia.

3