Credible Forward Guidance∗
Takeki Sunakawa‡
Hitotsubashi University

Taisuke Nakata†
University of Tokyo

First Draft: November 2018
This Draft: April 2022

Abstract
How can the central bank credibly implement a “lower-for-longer” strategy? To answer thisquestion, we analyze a series of optimal sustainable policy problems—indexed by the durationof reputational loss—in a sticky-price model with an effective lower bound (ELB) constraint
on nominal interest rates. We find that, even without an explicit commitment technology,
the central bank can still credibly keep the policy rate at the ELB for an extended period—
though not as extended as under the optimal commitment policy—and meaningfully mitigate
the adverse effects of the ELB constraint on economic activity.

JEL: E32, E52, E61, E62, E63
Keywords: Average Inflation Targeting, Effective Lower Bound, Forward Guidance, Sustainable Plan, Time-Consistency.

∗
We thank Roberto Billi, Timothy Hills, Narayana Kocherlakota, Kiminori Matsuyama, Sebastian Schmidt,
Francesco Zanetti, and seminar participants at the 2019 Workshop of the Australasian Macroeconomics Society (WAMS), the 2020 World Congress, the 5th End-of-Year Conference held at the Canon Institute of
Global Studies, the Forward Guidance Workshop 2021 Oxford and MMF, VEAMS, Asian Development Bank
Institute, Bank of Japan, Board of Governors of the Federal Reserve System, Central Bank of Peru, Hitotsubashi University, Hosei University, Kobe University, National Graduate Institute for Policy Studies, National
University of Singapore, and University of Tokyo for comments and suggestions. We also thank Katherine
Arnold, Quentin Batista, and Philip Coyle for their editorial assistance. Taisuke Nakata is supported by JSPS
Grant-in-Aid for Scientific Research (KAKENHI), Project Number 19K24682. Takeki Sunakawa is supported
by JSPS Grant-in-Aid for Scientific Research (KAKENHI) for Young Scientists, Project Number 18K12743.
†
Faculty of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033; Email:taisuke.nakata@e.u-tokyo.ac.jp‡
Faculty of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, 186-8601; Email:t.sunakawa@r.hit-u.ac.jp1

1

Introduction
In order to anchor longer-term inflation expectations at this level, the Committee
seeks to achieve inflation that averages 2 percent over time, and therefore judges
that, following periods when inflation has been running persistently below 2 percent,appropriate monetary policy will likely aim to achieve inflation moderately above
2 percent for some time.
Board of Governors of the Federal Reserve System 1
To maintain the symmetry of its inflation target, the Governing Council recognises
the importance of taking into account the implications of the effective lower bound.
In particular, when the economy is close to the lower bound, this requires especiallyforceful or persistent monetary policy measures to avoid negative deviations fromthe inflation target becoming entrenched. This may also imply a transitory period
in which inflation is moderately above target.
European Central Bank 2
Developing effective strategies to manage the adverse consequences of the effective lower

bound (ELB) constraint on nominal interest rates is an important task for economists andcentral bankers. In forward-looking models with an ELB, the commitment to keeping thepolicy rate at the ELB for an extended period—and temporarily overshooting inflation andoutput targets—is known to be effective in stimulating economic activity during a deep recession, as the anticipation of an overheated economy leads forward-looking households and firms
to increase consumption and set higher prices.3 We refer to this type of policy as overheating
commitment or lower-for-longer policy in this paper.
Recently, as shown in the quotes above, some central banks such as the Federal Reserve
Board and the European Central Bank have become more willing to allow temporary overshooting of inflation above the target. (The Bank of Japan has also adopted this type ofpolicy since November 2016.) While these policies cannot be strictly interpreted as a form ofcommitment, they share common elements with overheating commitment or lower-for-longer
policy.
One key argument against overheating commitment policy is its potential time-inconsistency.
Ex ante, it is desirable to promise to overheat the economy in the future, as the expectationsof future overheating stimulate inflation and output when the economy faces headwinds and1“2020 Statement on Longer-Run Goals and Monetary Policy Strategy” https://www.federalreserve.
gov/monetarypolicy/review-of-monetary-policy-strategy-tools-and-communications-statement-onlonger-run-goals-monetary-policy-strategy.htm.
2“The ECB’s monetary policy strategy statement” https://www.ecb.europa.eu/home/search/review/
html/ecb.strategyreview_monpol_strategy_statement.en.html.
3See, for example, Reifschneider and Williams (2000); Eggertsson and Woodford (2003); Jung, Teranishi,
and Watanabe (2005); Adam and Billi (2006).

2the ELB is a binding constraint. However, once the headwinds dissipate, the central bankwill have an incentive to renege on the promise of overheating the economy by raising thepolicy rate, because the overheating is ex post undesirable. Then, a natural question would
be how much overheating is needed to stimulate the economy while keeping monetary policy
credible.
In this paper, we study credible overheating commitment policies in a sticky-price modelwith the ELB with an eye towards understanding the best allocations the central bank cancredibly achieve when the optimal commitment policy is not credible. Specifically, we formulate and solve a series of optimal sustainable policy problems in which the central bank chooses
state-contingent allocations to maximize welfare subject to not only private-sector equilibrium
conditions, but also an incentive compatibility constraint—known as the sustainability constraint. The sustainability constraint requires that the continuation value associated withthe chosen state-contingent allocation has to be at least as large as the continuation value
associated with deviating from that allocation—and falling into a discretionary regime for N
periods—at any time and after any history of shocks. Under certain conditions discussed in
Nakata (2018), the sustainability constraint does not bind and the optimal sustainable policy
coincides with the optimal commitment policy.4 Our main interest is to characterize optimal
sustainable policies when the sustainability constraint occasionally binds.
Our main result is that, even when the optimal commitment policy is not credible, thecentral bank can still credibly keep the policy rate at the ELB for an extended period in
the aftermath of a crisis—though not as extended as under optimal commitment policy. As
in optimal commitment policy, such lower-for-longer policy generates a temporary post-crisis
overheating of the economy and mitigates the declines in output and inflation in a crisis
through expectations. Under reasonable assumptions regarding how long the central banksuffers from a loss of reputation after reneging on the promise of lower-for-longer, the welfare
cost of the ELB constraint is substantially lower under an optimal sustainable policy thanunder optimal discretionary policy.
One key feature of optimal sustainable policies is that they are less history dependent than
optimal commitment policy. As discussed in detail by Eggertsson and Woodford (2003), a
key feature of optimal commitment policy is history dependence. In particular, under optimalcommitment policy, the additional period at which to keep the policy rate at the ELB in theaftermath of a crisis—as well as the magnitude of output and inflation overshoot—increases
as the realized crisis shock duration increases. When the reputational force is strong, optimal
4
Specifically, in Nakata (2018), if the central bank were to renege on the promise of overheating the
economy in the aftermath of a crisis, it would lose reputation and private-sector agents would not believe
similar promises in future crises. If private-sector agents do not believe the central bank’s promise to overheat
the economy, future ELB episodes will be associated with large declines in inflation and output. Thus, concern
for maintaining reputation gives the central bank an incentive to fulfill the promise of keeping the lower-forlonger promise. According to Nakata (2018), this incentive to maintain reputation dominates the short-run
incentive to eliminate the overheating of the economy—and as a result, the optimal commitment policy iscredible—if that the policy rate is expected to fall into the ELB in the future with sufficient frequency andthe loss of reputation lasts for a sufficiently long duration.

3

sustainable policies exhibit qualitatively similar history dependence, though the degree of history dependence is weaker than under optimal commitment policy. When the reputational
force is sufficiently weak, optimal sustainable policies do not feature any history dependence.
That is, the additional ELB duration as well as the magnitude of output and inflation overshoot do not depend on the realized crisis shock duration.
We also analyze similarities between optimal sustainable policies and average inflation
targeting policies. Average inflation targeting can be seen as a way of implementing overheating commitment policies (see Budianto, Nakata, and Schmidt, 2020). When the averaging
window is one period, average inflation targeting coincides with the standard inflation targeting, which acts similarly to the optimal discretionary policy. When the averaging window
is infinity, average inflation targeting coincides with the price-level targeting, which acts similarly to the optimal commitment policy. With the averaging window in between one andinfinity, average inflation targeting can capture a variety of outcomes with differing degrees
of commitment, just as optimal sustainable policies can capture a variety of outcomes withthe punishment duration. Therefore, our optimal sustainable policies can be interpreted asoffering a microfoundation of average inflation targeting policies.
Our optimal sustainable policies are of interest to central banks for two reasons. First,
by construction, the optimal sustainable policies are time-consistent; thus, it is immune tothe criticism that the promised overshoot of inflation and output associated with any lowerfor-longer strategies may not be credible. Second, when the duration of reputational loss issufficiently short, optimal sustainable policies are not history dependent or not as history
dependent as optimal commitment policy is. Thus, it overcomes the criticism that, because
the policy rate path associated with a lower-for-longer strategy is complex, it is difficult for
central banks to clearly explain these strategies to the public.
Our paper builds on the literature on optimal monetary policy in the New Keynesianmodel with the ELB. This literature has demonstrated the effectiveness of lower-for-longer
policies in stimulating the economy at the ELB, assuming that the central bank is equippedwith an explicit commitment technology (Eggertsson and Woodford (2003); Jung, Teranishi,
and Watanabe (2005); Adam and Billi (2006); and Nakov (2008)). Our paper contributes
to this body of work by characterizing optimal sustainable policies in a model with the ELB
and showing that the central bank can credibly engage in lower-for-longer policies even in theabsence of an explicit commitment technology.
Within the literature on optimal policy and the ELB, some authors have explored ways toimplement lower-for-longer policies at the ELB in a time-consistent way. Eggertsson (2006)
and Burgert and Schmidt (2014) show that in models with non-Ricardian fiscal policy, a
discretionary government can provide incentives to a future government to keep the policy
rate at the ELB for longer by adopting expansionary fiscal policy and raising the nominal levelof government debt. Jeanne and Svensson (2007), Berriel and Mendes (2015), and Bhattarai,
Eggertsson, and Gafarov (2013) show that central banks’ balance sheet policies can act as4

a commitment device that allows the central bank to credibly implement lower-for-longer
policies. Billi (2017) and Nakata and Schmidt (2019b) analyze policy delegation in models
with the ELB, showing that lower-for-longer policies can be implemented in a time-consistent
way if the discretionary central bank’s standard dual-mandate objective function is replaced
by a nominal-income stabilization objective or augmented with an interest-rate smoothing
objective, respectively. Unlike these papers that either introduce a new policy instrument or
modify the central bank’s objective function, we use reputation to achieve lower-for-longer
policies in a time-consistent way.
Our paper is closely related to Nakata (2018) and Walsh (2018). Nakata (2018) has
shown that optimal commitment policy in the New Keynesian model with the ELB can bemade time-consistent by a particular trigger strategy capturing the reputational concern of thecentral bank. Our paper is different from Nakata (2018) because we study the best allocations
the central bank can credibly achieve when the optimal commitment policy is not credible,
whereas Nakata (2018) characterizes the conditions under which the optimal commitment
policy is credible. Walsh (2018) examines credibility of simple policy rules with forward
guidance—those that keeps the policy rate at the ELB for a fixed number of periods aftercrises—and reaches a conclusion similar to that of Nakata (2018).5 Our paper is different fromWalsh (2018) because we characterize the optimal allocation the central bank can credibly
achieve subject to a sustainability constraint, whereas Walsh (2018) studies credibility ofsimple policy rules that may or may not be optimal.6 It turns out that there is an interesting
relationship between our optimal sustainable policies and the forward guidance policy of Walsh(2018), which will be discussed in detail in Section 4.1.7
This paper is also closely related to the work of Dong and Young (2019). Using therecursive method of Abreu, Pearce, and Stacchetti (1990), Chang (1998), and Phelan andStacchetti (2001), Dong and Young (2019) characterize the entire set of sustainable plans
in a fully nonlinear New Keynesian model with the ELB. They find that the commitment
outcome is not sustainable in their model and that the central bank does not lower the policy
rate to the ELB under the best sustainable plan. Our paper is methodologically different
from theirs because we characterize a particular (countable) subset of sustainable plans thathave a trigger structure and in which on-equilibrium paths are given by the solutions to asequence of optimal sustainable policy problems indexed by the duration of reputational loss.
Finally, our paper is related to a set of papers that characterize optimal allocations in
5

See also Sukeda (2018), which extends the analysis of Walsh (2018) to a model with a discounted Euler
equation and a discounted Phillips curve.
6
Nakata (2018) and Walsh (2018) in turn build on earlier work of reputation in macroeconomics, including
Barro and Gordon (1983), Rogoff (1987), Chari and Kehoe (1990), Chang (1998), and Phelan and Stacchetti(2001), among many others. Recent contributions include Kurozumi (2008) and Loisel (2008).
7
See also Barthélemy and Mengus (2018) who examine sustainability of optimal commitment policy ina model with the ELB constraint in which the central bank’s objective function—either a benevolent orconservative kind—is unknown to private-sector agents and there is an inflationary bias. In their model, thebenevolent central bank can make the optimal commitment policy sustainable by raising inflation prior to aliquidity trap and signaling its type to private-sector agents.

5

macroeconomic models with a sustainability constraint. Kehoe and Perri (2002) characterize
the optimal allocation in an international business cycle model in which a deviation from thepromised plan would push the economy to autarky. Fujiwara, Kam, and Sunakawa (2019b)
study the optimal sustainable policy in a two-country model in which the deviation from thepromised cooperative plan would push the countries into a non-cooperative regime. Most
closely related to our paper is Sunakawa (2015) who characterizes the optimal sustainable
policy in a New Keynesian model with cost-push shocks but without the ELB constraint.
Our paper applies the same analytical framework and methodology used in these papers tomodels with the ELB.
The rest of the paper is organized as follows. Section 2 describes the model and the central
bank’s optimization problems. Section 3 presents the results. Section 4 provides additional
discussions and results. Section 5 concludes.

2

Model

2.1

Private sector

Our main model is a semi-loglinear New Keynesian model with a static Phillips curve.
The private-sector equilibrium conditions of this model are given by:

yt (st ) = Mee Et yt+1 (st+1 ) − σ(it (st ) − Et πt+1 (st+1 ) − r∗ ) + st(1)

πt (st ) = κyt (st )(2)

it (st ) ≥ iELB

(3)

where yt is output, πt is inflation, and it is the policy rate. Equations (1) and (2) are the Euler
equation and the static Phillips curve, respectively. Inequality (3) imposes the ELB constraint,
denoted by iELB , on the policy rate. σ is the intertemporal elasticity of substitution, r∗ > 0
is the natural rate of interest at the deterministic steady state, and κ is the slope of the static
Phillips curve. Mee ≤ 1.0 is a parameter whose value is less than one to diminish the degree of
forward-lookingness in the case of a discounted Euler equation.8 st is a natural rate shock. stdenotes a history of shocks up to time t. That is, st := {sk }tk=1 . Because there is uncertainty,
allocations are state-contingent and depend on st . We refer to the state-contingent sequence ofconsumption, inflation, and the nominal interest rate, {yt (st ), πt (st ), it (st )}∞t=1 , as an outcome.
Given a process for st , an outcome is said to be competitive if, for all t ≥ 1 and st ∈ St , (i)yt (st ) ∈ R, πt (st ) ∈ R, it (st ) ∈ R, where R denotes a set of real numbers, and (ii) equations(1)-(3) are satisfied.
We assume that st follows a two-state Markov process. st = r∗ > 0 in the “high” or
8In the baseline model, we set Mee = 1. We also consider a model with discounted Euler equation by
setting Mee ≤ 1 in Appedix F.

6

“normal” state, whereas st = rc < 0 in the “low” or “crisis” state. The probability of movingfrom the high/normal state to the low/crisis state is denoted by pH and will be referred toas the crisis frequency, whereas the probability of moving from the low/crisis state to the
low/crisis state is denoted by pL and will be referred to as the crisis persistence. Following
Nakata (2018), we allow pH to be non-zero, which opens up the possibility for a reputational
concern to make lower-for-longer policies credible.
The central bank’s value at period t is given byt

Vt (s ) := Et∞Xβ j u πt+j (st+j ), yt+j (st+j )

(4)

j=0

where the per-period objective function is given by the following function.
u(π, y) := −i1h 2π + λy 22(5)This quadratic objective function can be obtained as the second-order approximation to thehousehold’s welfare.9 For any outcome, there is an associated state-contingent sequence ofvalues, {Vt (st )}∞t=1 , which will be referred to as the value sequence.
We use this model with a static Phillips curve as our baseline model for a computational
reason. As described in Section 2.4, we use a time-iteration method—a commonly usednumerical method for nonlinear models—to solve our model. Regardless of the specificationof the Phillips curve, this solution method fails to converge if the duration of reputationalloss, N —a key parameter governing how long the central bank is prohibited from engaging
in state-contingent policies after it reneges on a previously announced policy rate path—issufficiently short. As discussed in Appendix E, we can devise alternative solution methods if
the duration of reputational loss is sufficiently short in the model with a static Phillips curve,
while we cannot do so in the model with a forward-looking Phillips curve.10 At the end of theday, we can solve the model with a static Phillips curve for any durations of reputational loss,
while we cannot do so in the model with a forward-looking Phillips curve.11 In Section 4.3,
we present some selected results from the model with a forward-looking Phillips curve andconfirm that key features of the optimal sustainable policy in the model with the static Phillips
curve survive in the model with a forward-looking Phillips curve.12

9See, for example, Woodford (2003) and Galí (2015).
We exploit the fact that the dynamics of the economy are history independent when N is sufficiently
small. See also Section 3.2 and Appendix D.
11
The instability of the time-iteration method in the presence of a sustainability constraint with low valuesof N is not specific to the model with ELB. The second author of this paper experienced a similar instability
issue in Fujiwara, Kam, and Sunakawa (2019a). We leave the task of developing a robust algorithm for models
with sustainability constraints to future research.
12
See also Bilbiie (2019) who uses a model with a static Phillips curve. See Nakata, Ogaki, Schmidt, andYoo (2019) and Levin and Sinha (2020a,b) for a systematic analysis of optimal commitment policies withvarying degrees of discounting in the Euler equation and Phillips curve.
10

7

2.2

Central bank

We will consider three classes of competitive outcomes that differ in how the central banksets its interest rate policy: the discretionary outcome, the commitment outcome, and thesustainable outcomes. Appendix B explains the optimization problem the central bank faces
in detail.
2.2.1

Discretionary outcome

At each time t, the discretionary central bank’s optimization problem is to choose {yt ,πt , it } to maximize the value today, taking as given the value function (Wt+1 (·)) and policy
functions for inflation and output (πt+1 (·) and yt+1 (·)) in the next period. That is,
Wt (st ) = max

πt ,yt ,itu(yt , πt ) + βEt Wt+1 (st+1 ),(6)

subject to equations (1), (2), and (3).
Let {Wd (·), πd (·), yd (·), id (·)} be the set of time-invariant value and policy functions thatsolve the Bellman equation above and in which the ELB binds only in the crisis state.13
They are functions of today’s shock realization, st . The discretionary outcome is definedas, and denoted by, the state-contingent sequence of output, inflation, and the policy rate,ttt{yd,t (st ), πd,t (st ), id,t (st )}∞t=1 such that yd,t (s ) = yd (st ), πd,t (s ) = πd (st ), and id,t (s ) = id (st )and the discretionary value sequence is defined as, and denoted by, {Vd,t (st )}∞t=1 such that
Vd,t (st ) = Wd (st ). We will also refer to the discretionary outcome as the outcome under the
optimal discretionary policy (ODP).
2.2.2

Commitment outcome

At the beginning of time one, the central bank with commitment technology chooses astate-contingent allocation, {yt (st ), πt (st ), it (st )}∞t=1 , to maximize the time-one value. Thatis,Vc,1 (s1 ) =max{yt (st ),πt (st ),it (st )}∞
t=1

E1

∞
X

β t−1 u(yt (st ), πt (st )),

(7)

t=1

subject to equations (1), (2), and (3) for all t ≥ 1 and after all histories of shocks st . The
commitment outcome, or the Ramsey outcome, is defined as the solution to this optimization
problem. In other words, the commitment outcome is a competitive outcome with the highest
time-one value. We denote the commitment outcome by {yc,t (st ), πc,t (st ), ic,t (st )}∞t=1 . The
value sequence associated with the commitment outcome is denoted by {Vc,t (st )}∞t=1 and will

13

There also exists a time-invariant solution to this discretionary government’s problem in which the ELBbinds in both states. See Armenter (2017), Nakata (2018), and Nakata and Schmidt (2019a) for extensive
analyses of such deflationary Markov-perfect equilibrium.

8be referred to as the commitment value sequence. We will also refer to the commitment
outcome as the outcome under the optimal commitment policy (OCP).
2.2.3

Sustainable outcomes

At the beginning of time one, the central bank chooses a state-contingent allocation,{yt (st ), πt (st ), it (st )}∞t=1 , to maximize the time-one value:
Vs,1 (s1 ) =max{yt (st ),πt (st ),it (st )}∞
t=1

E1

∞
X

β t−1 u(yt , πt ),

(8)

t=1

subject to equations (1), (2), and (3), and the following sustainability constraint,Et∞X

β k u(yt+k (st+k ), πt+k (st+k )) ≥ WdN (st ),

(9)

k=0

for all t ≥ 1 and after all histories of shocks, st . The left-hand side of the sustainability
constraint is the continuation value of implementing a chosen state-contingent allocation attime t after st . The right-hand side, WdN (st ), is the continuation value if the central bankdeviates from the chosen state-contingent allocation, with N indicating how many periods
it takes for the central bank to restore its lost reputation (“punishment” duration). During
the periods of reputational loss, the central bank cannot engage in state-contingent policies.
That is, the central bank has to act under discretion.
WdN (st ) is recursively defined as follows. For N = 0,
Wd0 (s) := Vs,1 (s),πd0 (s) := πs,1 (s),yd0 (s) := ys,1 (s).

In this case with N = 0, the punishment duration is zero and the central bank is not allowed
to deviate from the sustainable outcome. For any N > 0,
WdN (s) = max
π,y,iu(π, y) + βE[WdN −1 (s0 )|s]

where the maximization is subject to the private-sector equilibrium conditions, taking as given
the value and policy functions for the next period (that is, WdN −1 (·), πdN −1 (·), and ydN −1 (·)).
Note that the sustainability constraint has to be respected each period and for each historyof shocks, just as the Euler equation, the Phillips curve, and the ELB constraint have to berespected each period and for each history of shock. The sustainable outcome with N -period
reputational loss is defined as the solution to this infinite-horizon optimization problem. We
will also refer to the sustainable outcome with N -period reputational loss as the outcome
under the optimal sustainable policy (OSP) with N -period reputational loss.
Note that the punishment value, WdN (st ), is determined jointly with the sustainable outcome, except when N = ∞. When N = ∞, the punishment lasts forever and its value is9

given by the discretionary value, Wd (s), which is independent of the sustainable outcome. For
any finite N , the central bank eventually restores its reputation and the economy returns to
the allocations consistent with the sustainable outcome. Thus, the punishment value and thesustainable outcome are not independent of each other. All else equal, an increase (decrease)in the value associated with the sustainable outcome implies an increase (decrease) in thepunishment value.
As described in detail in Appendix A, once the sustainable outcome is computed from theoptimization problem above, we can construct a plan—a pair of central bank and privatesector strategies—that induces the sustainable outcome and that has a trigger-type structure.
In particular, we can construct a revert-to-discretion plan in which (i) the economy follows
the sustainable outcome as long as the central bank has never deviated from the policy rate
path consistent with the sustainable outcome in the past, and (ii) the economy follows thediscretionary outcome, or a temporary deviation to a discretionary regime, otherwise. By
construction, such a revert-to-discretion plan is credible, meaning that neither the central
bank nor private-sector agents have incentives to deviate from the instructions given by thestrategies. The central bank does not have an incentive to deviate from the policy rate
path consistent with the sustainable outcome because the sustainability constraint is imposedon the central bank’s optimization problem, ensuring that the continuation value under thesustainable outcome is at least as large as the punishment continuation value. Private-sector
agents do not have incentives to deviate from the private-sector strategy because the Euler
equation and the Phillips curves are satisfied, meaning that the output and inflation are
consistent with their optimizing behaviors given the central bank strategy. Even though
the deviation does not occur in equilibrium, the specification of what would happen if thecentral bank were to deviate from the sustainable outcome does affect what happens under
the sustainable outcome.
If the sustainability constraint does not bind at any time t and after any histories of shocks,
the sustainable outcome coincides with the commitment outcome. Also, if the sustainability
constraint always binds—which happens, for example, when the punishment length (N ) iszero or when the crisis frequency (pH ) is zero—the sustainable outcomes coincides with thediscretionary outcome. Our main interest is those cases in which the sustainability constraint
occasionally binds.

2.3

Parameter values

Table 1 shows the baseline parameter values. The quarterly frequency of crises is set to0.5/100 (=2/400). This choice is motivated by the fact that, in the United States, there
have been two large crises that pushed the short-term nominal interest rate to the ELB overroughly the last 100 years (400 quarters) since the creation of the Federal Reserve System.
The crisis shock persistence is set to 3/4, which implies the expected duration of the crisis
shock of 4 quarters. σ is set to 1. rc is chosen so that output declines 7 percent in the crisis
10state under the optimal discretionary policy. Conditional on the value of rc , the value of κ ischosen so that inflation declines 1 percentage point (annualized) in the crisis state under the
optimal discretionary policy. This severity of the crisis is consistent with that considered in
Boneva, Braun, and Waki (2016) and Nakata (2018), and is intended to capture the severityof the Great Recession of 2007-2009 in the United States.
Table 1: Baseline Parameter Values
Parameter
β
σ
κ
iELB
pH
pL
r∗
rc

Description
Discount factor
Intertemporal elasticity of substitution
Slope of the Phillips Curve
Effective lower bound on the policy rate
Frequency of the crisis state
Persistence of the crisis state
Natural rate in the normal state
Natural rate in the crisis stateN

Duration of reputational loss

Values
0.99251
0.25/700.5/100
3/4
3/400**Chosen so thatyd (st = rc ) = −0.07[20, 60, ∞]We consider three values for the duration of reputational loss (20, 60, and ∞), which arechosen to cover qualitatively distinct cases that can arise. To put these values into perspective,
note that the assumption that the central bank can restore its reputation after a finite numberof periods can be motivated by the fact that the tenure of governorship at central banks isfinite as well as the possibility that reputation may be specific to the leader of the central
bank, as opposed to the institution. As shown in Appendix H, the average tenure of thegovernorship in central banks in economies that have recently faced, or are currently facing,the ELB ranges from about 5 years (20 quarters) in the Bank of Japan to about 10 years(40 quarters) for the Bank of Canada. The maximum tenure duration exceeds 15 years (60quarters) at several central banks (the Federal Reserve, Bank of Canada, Bank of England,
and Sveriges Riksbank).

2.4

Solution method

The model is highly nonlinear, featuring two inequality constraints—the ELB constraint
and the sustainability constraint—and cannot be solved analytically. Following Kehoe andPerri (2002) and Sunakawa (2015), we recursify the infinite-horizon optimization problem of
the central bank into a saddle-point functional equation using the Lagrange multiplier on
the Euler equation as a pseudo-state variable. We then apply a time-iteration method tofind the set of time-invariant policy functions that solve the saddle-point functional equation.
Appendix C describes the details of the solution method as well as its accuracy.14
14

As discussed earlier, the time-iteration method fails to converge when N is sufficiently small. For thosesmall values of N under which the time-iteration method fails, we use alternative solution methods describedin Appendix E.

11

3Results

3.1

Dynamics

Figure 1 shows the dynamics of the economy under the ODP, the OCP, and OSPs with
N = [20, 60, ∞].15 In this figure, the crisis shock hits the economy at time 1 and stays there
until time 8. The crisis shock disappears at time 9 and the economy is in the normal statefrom that point on.
Figure 1: Dynamics
3

Policy Rate (ann. %)Output Gap (%)

Inflation (ann. %)

1

4
2

0.5

2
0
0
-2
1

OCP/OSP w/ N=

-4

-0.5

OSP w/ N=60
OSP w/ N=20

-6
0

1

10

20

ODP

1

10

20

-1

1

10

20

Note: ODP, OCP, and OSP stand for optimal discretionary policy, optimal commitment policy, and optimal
sustainable policy, respectively. The policy rate and the inflation rate are expressed in annualized percent.
The output gap is expressed in percent.

Under the ODP—shown by the solid red lines—the central bank keeps the policy rate
at the ELB as long as the crisis shock persists and raises the policy rate immediately after
the crisis shock disappears. Under the OCP—shown by the solid black lines—the central
bank keeps the policy rate at the ELB even after the crisis shock disappears, engineering theovershooting of inflation and output above their targets. Since households are forward looking,the anticipation of high inflation and high output in the aftermath of the crisis stimulates
economic activity during the crisis. The declines in inflation and output are substantiallysmaller under the OCP than under the ODP.
The allocations under the OSP with N = ∞ are identical to those under the OCP in thiscrisis scenario. As shown in the left panel of Figure 2, the value under the OSP with N = ∞—shown by the solid black—is always above the value in case the central bank deviates from the
OSP with N = ∞—shown by the dashed red line. That is, the sustainability constraint does
not bind. In our calibration, the crisis shock is sufficiently frequent so that the cost of being
unable to use lower-for-longer policies in the future forever outweighs the benefit of eliminating
the temporary overshooting of inflation and output targets. This result is consistent with the
15

Appendix D provides a detailed account of the dynamics of the economy, as well as an explanation why
the dynamics of the economy are history independent when N is sufficiently small under OSPs.

12

Figure 2: Value of fulfilling versus reneging on the promised allocations
Value
(N= )

0

Value
(N=60)

0

0

-100

-100

-100

-200

-200

-200

-300

-300

OSP
Reneging

-300
1

10

20

1

10

Value
(N=20)

20

1

10

20

Note: OSP stands for optimal sustainable policy.

finding of Nakata (2018) that a very small probability of being hit by the crisis shock suffices
to make the OCP credible.
When the loss of reputation is not as long, the cost of reneging on the lower-for-longer
promise in the aftermath of the crisis shock is smaller. In other words, the continuation value
in case of deviation is higher with a smaller N . The middle panel of Figure 2 shows thevalue of the OSP with N = 60 and the value of deviating from the OSP with N = 60 (solid
black and dashed red lines, respectively). According to the panel, the sustainability constraint
binds right after the crisis shock disappears, limiting the magnitude of the overshooting in
the aftermath of the crisis. The smaller overshoot means that inflation and output decline bymore during the crisis under the OSP with N = 60 than under the OCP and the OSP with
N = ∞. However, the declines in inflation and output are still much smaller under the OSP
with N = 60 than under the ODP. Similarly, the sustainability constraint binds right after
the crisis shock disappears under the OSP with N = 20, as can be seen in the right panelof Figure 2, limiting the magnitude of the overshoot. The overshoot in the aftermath of thecrisis is smaller—and as a result, the declines in inflation and output are larger—under the
OSP with N = 20 than under the OSP with N = 60. Even with N = 20, the declines in
inflation and output are still much smaller under the OSP than under the ODP.
Reflecting the less severe crisis under the OSP with N = 20 and N = 60, the welfare costof the ELB—shown in Table 2—is substantially lower under these OSPs than under the ODP.
With N = 60, welfare cost of the ELB is about 20 percent of that under the ODP and is only
slightly larger than under the OCP. Even with N = 20, welfare cost of the ELB is only about
half of the ODP.

13

Table 2: Welfare Cost of the ELB
Optimal commitment policy
Optimal sustainable policy
with N = ∞with N = 60
with N = 20
Optimal discretionary policy

abs(E[V ])29.5 (0.23)
29.5 (0.23)
34.6 (0.27)
63.3 (0.49)
128.1 (1.00)

Note: Numbers in parentheses are the welfare cost of the ELB relative to that under the optimal discretionary
policy.

3.2

History Dependence

Figure 3 shows the dynamics of output and the policy rate—displayed in the top andbottom rows, respectively—under three alternative realized durations of the crisis shock. The
first, second, and third columns are for the realized crisis shock duration of 1, 4, and 8 quarters,
respectively.
Figure 3: History Dependence (I)—Dynamics with Alternative Realized Crisis Shock Durations—5

Output Gap (%)(Shock Duration=1)

5

Output Gap (%)(Shock Duration=4)

5

0

0

0

-5

-5

-5

Output Gap (%)(Shock Duration=8)

OCP/OSP w/ N=
OSP w/ N=60
OSP w/ N=20
ODP

-10

3

1

10

20

Policy Rate (ann. %)(Shock Duration=1)

-10

3

1

10

20

Policy Rate (ann. %)(Shock Duration=4)

-10

3

2

2

2

1

1

1

0

1

10

20

0

1

10

20

0

1

10

20

Policy Rate (ann. %)(Shock Duration=8)

1

10

20

Note: ODP, OCP, and OSP stand for optimal discretionary policy, optimal commitment policy, and optimal
sustainable policy, respectively. The policy rate and the inflation rate are expressed in annualized percent.

14

Figure 4: History Dependence (II)8

Additional ELB Duration

Size of Output Gap
Overshooting

8

6

6

4

4

2

2

0

0
5

10

15

20

5

Realized Shock Duration

10

15

20

Realized Shock Duration
OCP
OSP w/ N=
OSP w/ N=60
OSP w/ N=20
ODP

Note: ODP, OCP, and OSP stand for optimal discretionary policy, optimal commitment policy, and optimal
sustainable policy, respectively. The output gap is expressed in percent.

Under the OCP—shown by the solid black lines—the additional ELB duration is 2, 4,
and 6 quarters when the realized crisis shock duration is 1, 4, and 8 quarters, respectively, as
can be seen in the bottom panels of Figure 3. The magnitude of the output overshoot is 2
percentage points, 4 percentage points, and 5 percentage points when the realized crisis shock
duration is 1, 4, and 8 quarters, respectively, as can be seen in the top panels of Figure 3.
Thus, both the additional ELB duration and the size of the inflation and output overshootdepend on the realized crisis shock duration. This dependence can be seen in Figure 4, whichshows how the additional ELB duration and the size of the output gap overshoot—shown in
the left and right panels, respectively—vary with the realized crisis shock duration.16
The history dependence of the OCP is in sharp contrast with the lack of history dependence
in the ODP. Under the ODP—shown by the solid red lines in Figure 3 and 4—the additional
ELB duration and the size of output gap overshoot do not depend on the realized crisis shock
duration: they are 0 quarter and 0 percent, regardless of the realized crisis shock duration.
OSPs with sufficiently large N s exhibit qualitatively similar history dependence to thatof the OCP. With N = ∞, the dynamics of the economy under the OSP—shown by thedashed lines in Figure 3 and 4—are identical to those under the OCP when the realized crisis
duration is sufficiently small. However, according to the right panel of Figure 4, when therealized crisis duration is sufficiently long—longer than 12 quarters—the size of the output
gap overshoot does not increase further with an increase in the realized crisis shock duration,
because the sustainability constraint binds and limits the magnitude of the output overshoot.
16

In computing the additional ELB duration and the size of the output gap overshoot, we assume that,prior to the crisis shock, the economy has been in the normal state for some time and the Lagrange multiplieron the Euler equation is zero in the period right before the crisis shock materializes.

15

Because the crisis persistence is 0.75, the probability that the crisis shock lasts for more than
12 quarters is very small, but it is not zero. Thus, the OSP with N = ∞ is less history
dependent than the OCP.
Under the OSP with N = 60, the additional ELB duration and the size of the output
gap overshoot increase with the realized crisis duration when the realized crisis durationis short, as can be seen by the dash-dotted lines in Figure 3 and 4. In particular, when therealized crisis duration is longer than 3 quarters, they do not increase further with the realizedcrisis duration, as the sustainability constraint binds and limits the magnitude of the output
overshoot. The OSPs with N = 60 is history dependent but is less history dependent thanthe OCP or the OSP with N = ∞.
OSPs are not history dependent at all when the duration of reputational loss is sufficiently
short. As can be seen by the dotted lines in Figure 3 and 4, with N = 20, the additional ELB
duration and the size of the output gap overshoot do not depend at all on the realized crisis
shock duration: they are 2 quarters and 2 percentage points. That is, the OSP with N = 20
is history independent.
Note that the magnitude of overshooting is slightly larger under the OSP with N = ∞than under the OCP when the realized crisis shock duration is 9 to 13 quarters—at or slightly
below the shortest crisis shock duration associated with the binding sustainability constraint
under the OSP with N = ∞. Consistent with this pattern, the post-crisis path of the policy
rate is slightly lower under the OSP with N = ∞ than under the OCP when the realizedcrisis shock duration is 9 to 13 quarters (not shown). In fact, the post-crisis ELB durationis one period longer under the OSP with N = ∞ than under the OCP when the realizedcrisis shock duration is 12 quarters. Similarly, the magnitude of overshooting is slightly larger
under the OSP with N = 60 than under the OCP or the OSP with N = ∞ when the realizedcrisis shock duration is 2 to 4 quarters—at or slightly below the lowest crisis shock duration
associated with the binding sustainability constraint under the OSP with N = 60.
How can the magnitude of overshooting be slightly larger with a smaller N for certain
realized crisis shock durations? Consider the following situation: the crisis shock has persisted
for a long time, and if the economy were to return to the normal state in the next period, themagnitude of post-crisis overshooting will not be limited by the sustainability constraint. In
this situation, suppose that the magnitude of post-crisis overshooting will be limited by thesustainability constraint down the road in case the crisis lasts for longer. Then, the possibilityof being constrained by the sustainability constraint down the road exerts downward pressureson output and inflation now through expectations. In this situation, because the sustainability
constraint will not be binding if the economy returns to the normal state in the next period,
the central bank can counteract that force by promising a larger post-crisis overshooting incase the economy were to return to the normal state in the next period—larger than what it
would promise if the sustainability constraint would not bind down the road.

16

3.3

On “reasonable” duration of reputational loss

When the loss of reputation lasts for a long time, the power of reputation is strong and
OSPs resemble the OCP. When the loss of reputation lasts for a short time, the power of
reputation is weak and OSPs resemble the ODP. A natural question that arises is what
reasonable values of the duration of reputational loss are.
One way to think about the reasonable duration of reputational loss is to hypotheticallyask how long it might take for a central bank to restore its reputation once it loses it. The
reasonable value of the duration of reputational loss based on this thought experiment may
depend on various factors: whether one believes the central bank’s reputation is individualspecific or institution-specific and the tenure duration of the central bank’s governors or chairs,
if one believes in the individual specific nature of reputation.
Another way to think about the reasonable duration of reputational loss is to theoreticallyrefine the concept of sustainability. In models in which the commitment and discretionary
outcomes are different, there are multiple—typically infinitely many—sustainable plans. In
this paper, we study infinitely many (countable) sustainable plans indexed by the durationof lost reputation. By imposing further restrictions on the set of sustainable plans and thus
refining the concept of sustainability, we can select one of these sustainable plans as being
more reasonable than others.
One refinement concept for any sequential equilibria—a sustainable plan in our setup—developed by game theorists in the context of two-player games is renegotiation proofness.
Roughly speaking, renegotiation proofness requires that, even if the deviation from an equilibrium were to occur hypothetically, two players would have no incentives to renegotiate thecontract—strategies in our setup—that they have initially agreed on. According to one definition of renegotiation-proofness proposed by Pearce (1987), a sustainable plan is renegotiationproof if the punishment value associated with that plan is higher than the punishment valueof any other sustainable plans.17
To apply this concept of renegotiation proofness to our model, the dash-dotted red line in
Figure 5 shows the punishment value associated with deviating from the OSPs with different
values for N . According to the figure, the punishment continuation value is non-monotonic.
When N is large, a reduction in the punishment duration increases the punishment value:
all else equal, it is good to stay in the discretionary regime for a shorter duration, as thevalue under the discretionary regime is lower than the value under the OSP. However, when
the punishment duration is sufficiently short and the sustainability constraint binds, a shorter
punishment duration lowers the value associated with the OSPs. This non-monotonicity arises
17

Farrell and Maskin (1989) proposed an alternative definition of renegotiation-proofness whereby a sustainable plan is renegotiation proof if there is no Pareto-improving move to another sustainable plan after
deviating from the on-equilibrium path at any point in time. As pointed out by Matsuyama (1997), in models with benevolent government where the government’s objective function and the private-sector’s objectivefunction coincide, this definition rules out any chance for the economy to achieve allocations better than the
ODP.

17

because a shorter punishment duration limits the size of the overshoot in the aftermath ofcrises and lowers the value associated with OSPs. As a result, when the punishment regime
ends, the economy will return to a sustainable outcome that is not as good as the sustainableoutcome with a longer punishment duration. When N is sufficiently small, this second effectdominates the first effect, and a shorter punishment duration lowers the punishment value.
Figure 5: Values in case of reneging
Normal-State Continuation Value

-50
-60
-70
-80
-90
-100
-110
-120

Sustainable Outcome
Reneging

-130
0

50

100

150

200

250

300

Duration of Reputation Loss

Note: Continuation value under the sustainable outcome after a crisis that has lasted for 8 quarters.

According to Figure 5, in our model, the sustainable outcome with the least severe punishment value is the sustainable outcome with N = 36. The dash-dotted red lines in Figure 6
show the dynamics of the economy under the OSP with N = 36, together with those under the OCP and the ODP. The post-crisis ELB duration is 3 quarters under the OSP with
N = 36, 3 quarters shorter than that under the OCP. The size of the post-crisis overshooting
is smaller—and output and inflation decline by more—under the OSP with N = 36 thanunder the OCP. However, the declines in inflation and output are much smaller under the
OSP with N = 36 than under the ODP. The welfare cost of the ELB constraint is about
one-third of that under the ODP.

4Additional results and discussion

4.1

Relation to the simple forward guidance policies of Walsh (2018)

We have shown that, when N is sufficiently small, the OSPs are not history dependent;
the policy rate path after the crisis shock disappears does not depend on the realized durationof the crisis shock. Thus, the OSPs bear some resemblance to the simple forward guidance
policies considered by Walsh (2018). Under the simple forward guidance policies of Walsh

18

Figure 6: Dynamics
3

Policy Rate (ann. %)Output Gap (%)

Inflation (ann. %)

1

4
2

0.5

2
0
0
-2
1
-4

OCP

-0.5

OSP w/ N=36

-6
0

1

10

20

ODP

1

10

20

-1

1

10

20

Note: ODP, OCP, and OSP stand for optimal discretionary policy, optimal commitment policy, and optimal
sustainable policy, respectively. The policy rate and the inflation rate are expressed in annualized percent.
The output gap is expressed in percent.

(2018), the central bank keeps the policy rate at the ELB for a fixed number of periods after
the crisis shock disappears, regardless of the realized duration of the crisis shock, and lets
the policy rate return to the steady-state level immediately thereafter. The only (minor)
difference is that, under the optimal sustainable policy, the policy rate does not return to thesteady state level immediately after liftoff. Instead, there is typically one period after liftoff
in which the policy rate is still below the steady-state level.
The similarity between the simple forward guidance policies and OSPs with small N s
points to one benefit of OSPs over the OCP; it may be easier for central banks to explainthese OSPs to the public than the OCP. One key criticism against the OCP is that it iscomplex. As Walsh (2018) argues, because of its complexity, it may be difficult for the central
bank in practice to steer the private-sector agents’ expectations in a way consistent with the
OCP. One dimension of complexity is history dependence. The OSPs have an advantage over
the OCP because they are less history dependent and thus simpler.
Note that, in our model, the ODP and the OSPs with small N s are history independent but
state-contingent. They are state-contingent because the policy rate path—in particular theliftoff date—depends on the realized crisis shock duration. Thus, these policies are different
from so-called calendar-based forward guidance that specifies the likely liftoff date, if that
guidance were to be narrowly or mistakenly interpreted as a non-state-contingent commitment
to raising the policy rate from the ELB at a particular date regardless of the evolution of theeconomy.18
18

Even when central banks indicate a likely date of liftoff from the ELB, they typically emphasize that theliftoff date will depend on the evolution of the economic outlook. That is, if the economy were to recover fasteror more slowly than in the baseline economic projection, the central bank will raise the policy rate from the
ELB earlier or later than the most likely liftoff date under the baseline projection. In practice, it is unlikelythat any central bank will ever engage in non-state-contingent forward guidance, though market participants
may not interpret the forward guidance specifying the likely date of liftoff as state-contingent as the central

19

4.2

Relation to the loose commitment approach of Bodenstein, Hebden,
and Nunes (2012)

Under OSPs, the central bank achieves crisis-state allocations that are “in between” that
under the ODP and that under the OCP. This feature of OSPs is reminiscent of the optimal policy obtained in a loose commitment approach in which the central bank reoptimizes
with a constant probability every period regardless of the incentive to renege on the priorcommitment.19 While these two approaches differ from each other in many ways, both approaches share the same spirit that they are intended to shed light on what the central bank
may be able to achieve when no explicit commitment technology is available. Indeed, recent
work by Fujiwara, Kam, and Sunakawa (2019a) shows that, when using a model without
the ELB, the allocations under the loose commitment approach with an appropriately chosen
re-optimization probability can approximate the allocation under the OSP with N -period
punishment reasonably well for any N . While we believe their result is likely to extend tothe model with ELB, it would be useful to verify the validity of their claim in our model infuture research.

4.3

Results from the model with a forward-looking Phillips curve

As discussed earlier, we have focused on the model with a static Phillips curve instead ofthe model with a forward-looking Phillips curve, because the range of durations of reputationalloss under which we can solve the model is wider in the model with a static Phillips curve.
A natural question is whether our key results thus far would extend to the model with aforward-looking Phillips curve. In this section, we discuss some select results from the modelwith a forward-looking Phillips curve.20
The private sector equilibrium conditions of the model with a forward-looking Phillips
curve are characterized by the Euler equation given by equation (1), the forward-looking
Phillips curve,
πt (st ) = κyt (st ) + βEt πt+1 (st+1 )(10)

and the ELB constraint given by inequality (3). The central bank’s objective function is thesame as that in the model with a static Phillips curve. The ODP, the OCP, and the OSPs aredefined in ways that are similar to how they are defined in the model with the static Phillips
curve. The parameter values used are shown in Table 3. The values for β, σ, pH , and sH are
the same as in the previous section. κ is set to 0.005. pL is set to 0.5, implying the expected
duration of the crisis state of 2 quarters. λ is set to 1/16, a value consistent with equal weightsbank intends.
19
See Bodenstein, Hebden, and Nunes (2012) for an analysis of optimal monetary policy under loose
commitment in the model with ELB. See Levin and Sinha (2020a,b) for a more recent treatment.
20
In addition, we also considered a model with a static Euler equation and a forward-looking Phillips curveand a model with a discounted Euler equation and a static Phillips curve. See Appendix F and G. The main
results of our paper also hold in these two models.

20

on the volatility of the output gap and the volatility of the annualized rate of inflation. A
high value of λ and a low value of pL increase a range of the duration of reputational loss
values under which we can solve the model. With these parameter values, we could solve themodel for N ≥ 75. We show the dynamics of the model under the OSPs with three values of
N = [80, 160, ∞].
Table 3: Parameter Values—Model with the Forward-Looking Phillips Curve
Parameter
β
σ
κ
λ
pH
pL
r∗
rc
N

Description
Discount rate
Intertemporal elasticity of substitution
Slope of the Phillips curve
Relative weight on output volatility
Crisis shock frequency
Crisis shock persistence
Natural rate in the normal state
Natural rate in the crisis state
Punishment length

Parameter Value
1
1+0.0075 ≈ 0.9925
1
0.005[1/16]
0.5/100
0.5
3/400−0.0125[80, 160, ∞]Figure 7 shows the dynamics of the economy under the OCP, the ODP, and OSPs withthe three values of N listed above.
The dynamics of the economy under the OCP and the ODP are consistent with thosein the model with a static Phillips curve as well as those in the existing studies. Under the
OCP—shown by the solid black lines—the central bank keeps the policy rate at the ELB
even after the crisis shock disappears and engineers the overshooting of inflation and output
above their targets. Under the ODP—shown by the solid red lines—the central bank keeps
the policy rate at the ELB as long as the crisis shock continues and raises the policy rate
immediately once the crisis shock disappears.
Under the OSPs—shown by the dashed, dash-dotted, and dotted black lines for N = ∞,N = 160, and N = 80, respectively—the central bank keeps the policy rate at the ELB after
the crisis shock disappears, but not as long as it would do under the OCP. The magnitudes
of inflation and output overshoots are smaller under the OSPs than under the OCP, with themagnitudes smaller when N is smaller. Consistent with the magnitudes of the overshoot in
the aftermath of the crisis, the paths of output and inflation during the crisis state are lower
when N is smaller. While the OSPs are not as stimulative as the OCP, they still substantiallyreduce the welfare cost of the ELB constraint relative to the ODP, which can be seen in
Table 4.
One key feature of OSPs in the model with a static Phillips curve is that they are lesshistory dependent than the OCP. To examine whether this feature holds in the model with aforward-looking Phillips curve, Figure 8 shows how the additional ELB duration, the size ofthe output gap overshoot, and the size of the inflation overshoot vary with the realized crisis
21

Figure 7: Dynamics—Model with Forward-Looking Phillips Curve—
4

Policy Rate (ann. %)Output Gap (%)
5

1

0

0.5

-5

0

Inflation (ann. %)

3
2
1
0

1

10

20

-10

1

10

20

-0.5

OCP
OSP w/ N=
OSP w/ N=160
OSP w/ N=80
ODP

1

10

20

Note: ODP, OCP, and OSP stand for optimal discretionary policy, optimal commitment policy, and optimal
sustainable policy, respectively. The policy rate and the inflation rate are expressed in annualized percent.
The output gap is expressed in percent.

Table 4: Welfare Cost of ELB—Model with Forward-Looking Phillips Curve—
Optimal commitment policy
Optimal sustainable policy
with N = ∞with N = 160
with N = 80
Optimal discretionary policy

abs(E[V ])26.8 (0.39)
27.1 (0.39)
28.0 (0.40)
29.9 (0.43)
68.9 (1.00)

Note: Numbers in parentheses are the welfare cost of the ELB relative to that under the optimal discretionary
policy.

shock duration. In the figure, the solid black and red lines are for the OCP and the ODP,
respectively, whereas the dashed, dash-dotted, and dotted black lines are for the OSPs with
N = ∞, N = 160, and N = 80, respectively. According to the figure, the additional ELB
duration, the size of the output gap overshoot, and the size of the inflation overshoot are lesssensitive to the realized crisis shock duration under the OSPs than under the OCP. That is,
the OSPs are less history dependent than the OCP. Also, the OSPs are less history dependent
with smaller N s.
All told, qualitatively, the key insights from the model with a static Phillips curve carryover to the model with a forward-looking Phillips curve.

4.4

Results from the model with average inflation targeting

In this section, we analyze similarities between optimal sustainable policies and average
inflation targeting (AIT) policies—policies that feature overheating commitment as in OSPs
22

Figure 8: History Dependence—Model with Forward-Looking Phillips Curve—

7

Additional
ELB Duration

Size ofOutput Gap Overshooting

Size of
Inflation Overshooting

6

0.2

6
0.15

5

4

4

0.1

3
2

2
0.05

1
0
2

4

6

8

Realized Shock Duration

0

2

4

6

8

Realized Shock Duration

0

2

4

6

8

Realized Shock Duration

OCP
OSP w/ N=
OSP w/ N=160
OSP w/ N=80
ODP

Note: ODP, OCP, and OSP stand for optimal discretionary policy, optimal commitment policy, and optimal
sustainable policy, respectively. The inflation rate are expressed in annualized percent. The output gap isexpressed in percent.

and that have recently attracted a lot of attentions recently because of new monetary policy
frameworks at the Fed and the ECB.
We consider a policy ruleit = st + φω −1 π̂t ,(11)

π̂t = ωπt + (1 − ω)π̂t−1 ,

(12)

where
φ is the degree of response to inflation, and ω ∈ [0, 1] is a weight on the current inflation rate
in an exponential moving average inflation rate π̂t . The central bank sets the policy rate byreferring to the natural rate of interest st as in Cúrdia, Ferrero, Ng, and Tambalotti (2015).
In addition, the central bank responds to an average of the past inflation rates π̂t . The rest of
the equilibrium conditions consist of the Euler equation and static Phillips curve, which are
given by Equations (1) and (2). We also consider the ELB constraint (3) as before.
This policy rule (11) and (12) nests the following different strategies (See also Budianto,
Nakata, and Schmidt, 2020):• When ω = 1, the Equation (11) becomes the standard Taylor ruleit = st + φπt .
23The central bank follows a standard flexible inflation targeting strategy. We assume
that the central bank knows the natural rate of interest st so that the standard inflation
targeting resembles the ODP.
• When ω = 0, the Equation (11) becomes
it = st + φ(pt − p−1 ).
The central bank aims to stabilize the price level pt ≡ πt + pt−1 .21 It is well known that
the OCP has such a property of price level targeting.22• When ω ∈ (0, 1), the central bank aims to stabilize an exponential moving average inflation rate π̂t as given in Equation (12). We are especially interested in these intermediatecases because they can be similar to OSPs which are also intermediate between the OCP
and the ODP.
The parameter values are set to the values in Table 1 on page 11.23 φ is set to 5.0 sothat the central bank responds to inflation aggressively. We have checked that our results arerobust to different values of φ.
Figure 9 shows the dynamics for different values of ω ∈ [0, 1] when the crisis shock hits
the economy at time 1 and remains there for 8 quarters as in the main exercise in Section
2 (see Figure 1). The dynamics under the AIT policy with ω = 0.01—shown by the solid
black lines—are similar to those under the OCP and the OSP with the punishment duration
N = ∞, whereas the dynamics under the AIT policy with ω = 1.0—shown by the solid
red lines—are similar to those under the ODP. As we increase the value of ω, the paths ofinflation, the output gap, and the policy rate diverge from those under OCP and converge tothose under ODP. This pattern is similar to what happens when we decrease N of OSPs in
Figure 1.
However, there are a couple of minor differences. First, in the left panel of Figure 9,
the additional duration of ELB under ω = 0.01 is shorter than the one under the OCP (see21

Note thatω −1 π̂t =∞X(1 − ω)j πt−j + (1 − ω)t ω −1 π̂−1 .
j=0Assuming π̂−1 = 0, we haveit =st + φ∞
X(1 − ω)j πt−jj=0

As ω → 0, we haveit = st + φ(pt − p−1 ).
22

Eggertsson and Woodford (2003) shows that, even in the presence of ELB, a modified price level targetingby setting the price level to an output gap-adjusted price index can mimic the OCP. The policy rule consideredhere only intends to set the price level constant and does not completely replicate the OCP.
23
With this value, we obtain yl = −7.0 under the ODP.

24

Figure 9: Dynamics under AIT policies
3

Policy Rate (ann. %)Output Gap (%)

Inflation (ann. %)

1

4
2
2

0.5

0
0
-2

1

-4

=0.01

-0.5

=0.15
=0.4

-6
0

=1.0

-1
1

10

20

1

10

20

1

10

20

Note: The policy rate and the inflation rate are expressed in annualized percent. The output gap is expressedin percent.

footnote 19). Second, the values of inflation, the output gap, and the policy rate in the normal
state after time 9 under the AIT policy with ω = 1.0 are lower than those under the ODP.
This implies that the uncertainty effect of ELB under AIT policies is different from that under
OSPs.24
Figure 10: History Dependence under AIT policies

8

Additional ELB Duration

8

6

6

4

4

2

2

0

5

10

15

0

20

Realized Shock Duration

Size ofOutput Gap Overshooting

5

10

15

20

Realized Shock Duration
=0.01
=0.05
=0.15
=0.4
=1.0

Note: The output gap is expressed in percent.

Figure 10 exhibits the additional ELB duration and the size of output gap overshooting
under AIT policies with different shock durations. The left panel of Figure 10 shows that,as the shock duration increases, the additional ELB duration also increases. However, theadditional ELB duration can be shorter under the AIT policies with different values of ω than24

For the uncertainty effect of ELB, see Nakata and Schmidt (2019a); Hills, Nakata, and Schmidt (2019).

25

under the OSPs with corresponding values of N .25 For example, when the shock duration is10, the additional ELB duration is 4 under ω = 0.01. In contrast, the same shock duration
leads to a longer additional ELB duration, 6, under the OCP as shown in Figure 4.
AIT policies are more history-dependent compared with OSPs. The AIT policy with
ω = 0.01 or 0.05 is history-dependent for the displayed range of the shock duration, which isless than 20. That is, in the right panel of Figure 10, the size of output gap overshooting isincreasing as the shock duration increases.26
The AIT policy with ω = 0.4 shows some degree of history independence. When the shock
duration is longer, the output-gap overshooting becomes less sensitive to the shock duration.
However, the OSP with N = 20 is completely history independent in Figure 4. That is, thesize of overshooting does not depend at all on the shock duration.

5

Conclusion
In this paper, we have characterized OSPs in models with the ELB constraint. We findthat, even when the OCP is not credible, the central bank can still credibly commit to
keeping the policy rate at the ELB in the aftermath of a crisis—though not as long as under
the OCP—and meaningfully mitigate the adverse consequences of the ELB constraint on
economic activity in crises.
By construction, our OSPs are time-consistent and thus overcome the criticism that thetemporary overheating of the economy associated with lower-for-longer strategies is not credible. When the loss of reputation is sufficiently short-lived, these OSPs are not history dependent or not as history dependent as the OCP. Thus, it overcomes the criticism that the
implied policy rate path is too complex for the central bank to be able to explain to the
public, making the OSPs even more attractive.
Although we focus on the time-consistency aspect of lower-for-longer policies in this paper,
there are other aspects of these policies that could make them less attractive in reality thanin theory. For example, the public may not understand the temporary nature of the inflation
overshooting, resulting in unanchoring of the long-run inflation expectations (Kohn (2012)
and Yellen (2018)). The overheating of the economy may be less desirable for policymakers inreality than what’s implied by our model if the overheating of the economy leads to financial
instability (Yellen (2018)). It would be useful to formally analyze how these factors affectthe effectiveness and implementability of lower-for-longer strategies. We leave such analysis
to future research.

25

One exception is the case of ω = 1.0, which corresponds to the ODP.
In general, there is a threshold value of shock duration above which the size of overshooting becomes
constant. This case displays that such a threshold is very large (at least more than 20) under the AIT policy
with ω = 0.01 or 0.05.
26

26

References
Abreu, D., D. Pearce, and E. Stacchetti (1990): “Toward a Theory of Discounted
Repeated Games with Imperfect Monitoring,” Econometrica, 58(5), 1041–1063.
Adam, K., and R. Billi (2006): “Optimal Monetary Policy Under Commitment with a
Zero Bound on Nominal Interest Rates,” Journal of Money, Credit, and Banking, 38(7),1877–1905.
Armenter, R. (2017): “The Perils of Nominal Targets,” Review of Economic Studies, 0,
1–37.
Barro, R., and D. Gordon (1983): “Rules, Discretion, and Reputation in a Model ofMonetary Policy,” Journal of Monetary Economics, 12, 101–121.
Barthélemy, J., and E. Mengus (2018): “The Signaling Effect of Raising Inflation,”
Journal of Economic Theory, 178, 488–516.
Bean, C. (2013): “Global Aspects of Unconventional Monetary Policies,” Remarks at the Federal Reserve Bank of Kansas City Economic Policy Symposium, Jackson Hole, Wyoming.
Berriel, T. C., and A. Mendes (2015): “Central Bank Balance Sheet, Liquidity Trap, andQuantitative Easing,” Mimeo.
Bhattarai, S., G. Eggertsson, and B. Gafarov (2013): “Time Consistency and the Duration of Government Debt: A Signalling Theory of Quantitative Easing,” NBER Working
Paper No. 21336.
Bilbiie, F. O. (2019): “Optimal Forward Guidance,” American Economic Journal: Macroeconomics, 11(4), 310–345.
Billi, R. (2017): “A Note on Nominal GDP Targeting and the Zero Lower Bound,” Macroeconomic Dynamics, 21(8), 2138–2157.
Bodenstein, M., J. Hebden, and R. Nunes (2012): “Imperfect Credibility and the Zero
Lower Bound,” Journal of Monetary Economics, 59(2), 135–149.
Boneva, L. M., A. R. Braun, and Y. Waki (2016): “Some Unpleasant Properties of
Loglinearized Solutions when the Nominal Rate is Zero,” Journal of Monetary Economics,
84, 216–232.
Budianto, F., T. Nakata, and S. Schmidt (2020): “Average Inflation Targeting and theInterest Rate Lower Bound,” CEPR Working Paper DP14400.
Bullard, J. (2013): “Monetary Policy in a Low Policy Rate Environment,” OMFIF Golden
Series Lecture, London, United Kingdom.
27

Burgert, M., and S. Schmidt (2014): “Dealing with a Liquidity Trap when Government
Debt Matters: Optimal Time-Consistent Monetary and Fiscal Policy,” Journal of Economic
Dynamics and Control, 47, 282–299.
Carney, M. (2012): “Guidance,” Remarks at the CFA Society Toronto, Toronto, Ontario.
Chang, R. (1998): “Credible Monetary Policies in an Infinite Horizon Model: Recursive
Approaches,” Journal of Economic Theory, 81, 431–461.
Chari, V. V., and P. Kehoe (1990): “Sustainable Plans,” Journal of Political Economy,
98(4), 783–802.
Clarida, R. H. (2019): “The Federal Reserve’s Review of Its Monetary Policy Strategy,
Tools, and Communication Practices,” Remarks at the 2019 U.S. Monetary Policy Forum,sponsored by the Initiative on Global Markets at the University of Chicago Booth School ofBusiness, New York City.
Cœuré, B. (2013): “The Usefulness of Forward Guidance,” Remarks at the Money Marketeers
Club of New York, New York City, New York.
Cúrdia, V., A. Ferrero, G. C. Ng, and A. Tambalotti (2015): “Has U.S. monetary
policy tracked the efficient interest rate?,” Journal of Monetary Economics, 70(C), 72–83.
Dong, B., and E. Young (2019): “Forward Guidance and Credible Monetary Policy,”
Mimeo.
Dudley, W. C. (2013): “Unconventional Monetary Policies and Central Bank Independence,” Remarks at the Central Bank Independence Conference: Progress and Challenges
in Mexico, Mexico City, Mexico.
Eggertsson, G. (2006): “The Deflation Bias and Committing to Being Irresponsible,” Journal of Money, Credit, and Banking, 38(2), 283–321.
Eggertsson, G., and M. Woodford (2003): “The Zero Bound on Interest Rates andOptimal Monetary Policy,” Brookings Papers on Economic Activity, 34(1), 139–235.
Farrell, J., and E. Maskin (1989): “Renegotiation in repeated games,” Games and Economic Behavior, 4, 327–360.
Fujiwara, I., T. Kam, and T. Sunakawa (2019a): “On Two Notions of Imperfect Credibility in Optimal Monetary Policies,” Economics Letters, 174, 22–25.
(2019b): “Sustainable International Monetary Policy Cooperation,” Journal of Economic Dynamics and Control, 106.
Galí, J. (2015): Monetary Policy. Inflation, and the Business Cycle. Princeton: Princeton
University Press.
28

George, E. L. (2019): “The Economic Outlook, Monetary Policy and the Federal Open
Market Committee’s Inflation Objective,” Remarks at the The Money Marketeers of New
York University, New York City.
Hills, T. S., T. Nakata, and S. Schmidt (2019): “Effective Lower Bound Risk,” European
Economic Review, 120, 103321.
Jeanne, O., and L. E. O. Svensson (2007): “Credible Commitment to Optimal Escapefrom a Liquidity Trap: The Role of the Balance Sheet of an Independent Central Bank,”
American Economic Review, 97(1), 474–490.
Jung, T., Y. Teranishi, and T. Watanabe (2005): “Optimal Monetary Policy at the
Zero-Interest-Rate Bound,” Journal of Money, Credit, and Banking, 35(7), 813–35.
Kehoe, P. J., and F. Perri (2002): “International Business Cycles with Endogenous Incomplete Markets,” Econometrica, 70(3), 907–928.
Kohn, D. L. (2012): “Interview with Donald Kohn,” Top of Mind, Goldman Sachs Research.
Kurozumi, T. (2008): “Optimal Sustainable Monetary Policy,” Journal of Monetary Economics, 55, 1277–1289.
Lacker, J. (2013): “Monetary Policy in the United States: The Risks Associated With
Unconventional Policies,” Remarks at the Swedbank Economic Outlook Seminar, Stockholm,
Sweden.
Levin, A. T., and A. Sinha (2020a): “Limitations on the Effectiveness of Forward Guidancein the context of the COVID-19 Pandemic,” NBER Working Paper 27748.
(2020b): “Pitfalls of Makeup Strategies for Mitigating the Effective Lower Bound,”Cato Journal, 40(2), 493–508.
Loisel, O. (2008): “Central bank reputation in a forward-looking model,” Journal of Economic Dynamics and Control, 11, 3718–3742.
Matsuyama, K. (1997): “Credibility and Intertemporal Consistency: A Note on Strategic
Macroeconomic Policy Models,” Macroeconomic Dynamics, 1, 658–665.
Nakata, T. (2018): “Reputation and Liquidity Traps,” Review of Economic Dynamics, 28,
252–268.
Nakata, T., R. Ogaki, S. Schmidt, and P. Yoo (2019): “Attenuating the forward guidance puzzle: Implications for optimal monetary policy,” Journal of Economic Dynamics
and Control, 106(C), 90–106.
Nakata, T., and S. Schmidt (2019a): “Conservatism and Liquidity Traps,” Journal of
Monetary Economics, 104, 37–47.
29

(2019b): “Gradualism and Liquidity Traps,” Review of Economic Dynamics, 31,
182–199.
Nakov, A. (2008): “Optimal and Simple Monetary Policy Rules with Zero Floor on the
Nominal Interest Rate,” International Journal of Central Banking, 4(2), 73–127.
Pearce, D. G. (1987): “Renegotiation Proof Equilibria: Collective Rationality and Intertemporal Cooperation,” Mimeo.
Phelan, C., and E. Stacchetti (2001): “Sequential Equilibria in a Ramsey Tax Model,”Econometrica, 69(6), 1491–1518.
Plosser, C. (2013): “Forward Guidance,” Remarks at Stanford Institute for Economic Policy
Research’s (SIEPR) Associates Meeting, Stanford, California.
Powell, J. H. (2019): “Monetary Policy: Normalization and the Road Ahead,” the 2019
SIEPR Economic Summit, Stanford Institute of Economic Policy Research, Stanford, California.
Reifschneider, D., and J. C. Williams (2000): “Three Lessons for Monetary Policy in a
Low-Inflation Era,” Journal of Money, Credit and Banking, 32(4), 936–966.
Rogoff, K. (1987): “Reputational Constraints on Monetary Policy,” Carnegie-Rochester
Conference Series on Public Policy, 26, 141–182.
Sukeda, A. (2018): “Credible Forward Guidance with Financial Disruption,” Mimeo.
Sunakawa, T. (2015): “A Quantitative Analysis of Optimal Sustainable Monetary Policies,”
Journal of Economic Dynamics and Control, 52, 119–135.
Ueda, K. (2013): “A Long and Bumpy Journey ahead for Forward Guidance,” in Forward
Guidance: Perspectives from Central Bankers, Scholars and Market Participants, ed. by
W. den Haan, pp. 165–170. VoxEU.org Ebook.
Walsh, C. (2018): “Simple Sustainable Forward Guidance at the ELB,” Mimeo.
Williams, J. C. (2012): “The Federal Reserve’s Unconventional Policies,” FRBSF Economic
Letter, 2012-34 (November 13).
Woodford, M. (2003): Interest and Prices: Foundations of a Theory of Monetary Policy.
Princeton: Princeton University Press.
Yellen, J. (2018): “Comments on Monetary Policy at the Effective Lower Bound,” Brookings
Papers on Economic Activity, Fall.

30

Technical Appendix for Online Publication
This technical appendix is organized as follows:• Appendix A defines some key concepts.
• Appendix B describes the equilibrium conditions characterizing the sustainable outcomein the model with a static Phillips curve in detail.
• Appendix C describes the numerical solution method and reports the solution accuracy.
• Appendix D analyzes the dynamics of the model with a static Phillips curve in detailthrough policy and value functions.
• Appendix E describes solution methods for the model with a static Phillips curve when
N is small.
• Appendix F describes additional results in the model with a static Phillips curve anddiscounted Euler equations.
• Appendix G describes the equilibrium conditions characterizing the sustainable outcomein the model with a static Euler equation in detail.
• Appendix H documents the average tenure of chairpersons/governors in select central
banks.
• Appendix I collects policymakers’ speeches in which the time-inconsistency problem of
the lower-for-longer policy is discussed.

A

Definition of a plan and credibility

This section defines a plan, credibility, and the revert-to-discretion plan. The definitions
closely follow Chang (1998) and Nakata (2018).

A.1

Plan

A government strategy, denoted by σg := {σg,t }∞
t=1 , is a sequence of functions that maps
a history of the nominal interest rates up to the previous period and a history of states up
to today into today’s nominal interest rate. Formally, σg,t is given by σg,1 : S → R≥0 andt
27 Given a particular realization of {s }∞ , a sequence of
σg,t : Rt−1t t=1
≥0 × S → R≥0 for all t ≥ 2.
nominal interest rates will be determined recursively by i1 = σg,1 (s1 ) and it = σg,t (it−1 , st ) for
all t > 1 and for all st ∈ St . A government strategy is said to induce a sequence of the nominal
27

The first period is a special case, as there is no previous policy action.

31

interest rates. A private-sector strategy, denoted by σp := {σp,t }∞
t=1 , is a sequence of functions
mapping a history of nominal interest rates up to today and a history of states up to today
into today’s consumption and inflation. Formally, σp,t is given by σp,t : Rt × St → (R, R) for
all t.
Given a government and private-sector strategy, a sequence of consumption and inflation
will be determined recursively by (yt , πt ) = σp,t (it , st ) for all t ≥ 1 and for all st ∈ St . A
private sector strategy, together with a government strategy, is said to induce a sequence ofconsumption and inflation.28 A plan is defined as a pair of government and private sector
strategies, (σg ,σp ). Notice that a plan induces an outcome—a state-contingent sequence ofconsumption, inflation, and the nominal interest rate. As discussed earlier, there is a value
sequence {wt (st )}∞t=1 , associated with any outcome.

A.2

Credibility

Let us use CEtI (s) to denote a set of state-contingent sequences of the nominal interest
rate consistent with the existence of a competitive equilibrium when st = s. Formally, foreach s ∈ S, CEtI (s) := {it (s) ∈ I∞ | ∃ (yt (s), πt (s)) s.t. (yt (s), πt (s), it (s)) ∈ CEt (s)}. σg issaid to be admissible if, after any history of policy actions, it−1 , and any history of states, st ,it (s) induced by the continuation of σg belongs to CEtI (st ).
A plan, (σg , σp ), is credible if (i) σg is admissible, (ii) after any history of policy actions, it ,and any history of states, st , the continuation of σp and σg induce a (yt (st ), πt (st ), rt (st )) ∈CEt (st ), and (iii) after any history it−1 and st , it (st ) induced by σg maximizes the government’s objective over CEtI (st ) given σp . In plain languages, a plan is said to be credible ifneither the private sector nor the government has incentive to deviate from the strategies
associated with it.
An outcome is said to be credible if there is a credible plan that induces it. When a certain
plan A is credible and the plan A induces a certain outcome α, we say that the outcome α
can be made credible, or time-consistent, by the plan A.

A.3

The revert-to-discretion plan

I now define a key object of this paper, the revert-to-discretion plan, and discuss thecondition under which this plan is credible.
The revert-to-discretion plan, (σgrtd , σprtd ), consists of (i) the following government strategy:
rtd
rtd (it−1 , st ) = i (st ) if i = i (sj ) for all j ≤ t − 1, andσg,1 = is,1 (s1 ) for any s1 ∈ S, σg,t
s,t
j
s,j
rtd (it−1 , st ) = i (st ) otherwise, and (ii) the following private-sector strategy: σ rtd (it , st ) =
σg,t
d,t
p,t
rtd (it , st ) = (y (s , i ), π (s , i )) otherwise,29(ys,t (st ), πc,t (st )) if rj = ic,j (sj ) for all j ≤ t, σp,t
br t t
br t twherehiybr (st , rt ) = Et yd,t+1 (st+1 ) − σ it − Et πd,t+1 (st+1 ) − r∗ + st
(13)
πbr (st , rt ) = κybr (st , rt ) + βEt πd,t+1 (st+1 )

(14)

28
Note that, while the nominal interest rate today depends on the history of nominal interest rates up to
the previous period, consumption and inflation today depend on the history of nominal interest rates up to
today. The implicit within-period-timing protocol behind this setup is that the government moves before theprivate sector does.
29
Subscript br stands for best r esponse.

32

The government strategy instructs the government to choose the nominal interest rate consistent with the sustainable outcome, but chooses the interest rate consistent with the discretionary outcome if it has deviated from the sustainable outcome at some point in the past. The
private sector strategy instructs the household and firms to choose consumption and inflationconsistent with the sustainable outcome as long as the government has never deviated fromthe sustainable outcome. If the government has ever deviated from the nominal interest rateconsistent with the sustainable outcome, the private sector strategy instructs the household
and firms to choose output and inflation today based on the belief that the government in the
future will choose the nominal interest rate consistent with the discretionary outcome. By
construction, the revert-to-discretion plan induces the sustainable outcome, and the impliedvalue sequence is identical to the sustainable value sequence.
It is relatively straightforward to show that the revert-to-discretion plan is credible. By
construction, Vs,t (st ) ≥ Vd,t (st ) for all t ≥ 1 and all st ∈ St , making sure that the government
does not have an incentive to deviate from the instruction given by the government strategy
after any history it−1 and st in which the optimal sustainable policy has been followed.
The revert-to-discretion plan that induces the optimal sustainable outcome with a finite
period punishment is defined in a similar way (see Nakata (2018) for rigorous exposition). It
is also straightforward to show that such a plan is credible.

B

Model with a static Phillips Curve
The policymaker maximizes∞
X

V0 = −E0

β t yt2 ,t=0

subject to
yt = (Mee + σ −1 κ)Et yt+1 − σ (it − r∗ ) + st ,(15)

it ≥ iELB ,∞X
2
Vt = −Et
β j yt+j≥ WdN (st ),

(16)
(17)

j=0

for all t ≥ 0, where we used a relationship based on the static Phillips curve, πt = κyt toobtain equation (15). The shock, st , follows two-state Markov
chain, st ∈ {sH , sL } where
1 − pH pH
sH > sL . Transition probability matrix is given as P =, where pH is the
1 − pL pL
frequency of the crisis and pL is the persistence of the crisis. WdN (st ) is the continuation value
under N -period punishment.
The analytical solution for the discretionary outcome: The ZLB is slack in thehigh/normal state, φH = 0, and the ZLB is binding in the low/crisis state, φL = φ > 0.

33

Thus, the equilibrium conditions become
yH = 0,yL = −φ,yH + σ(iH − r∗ − σ −1 sH ) − (Mee + σκ)(pH yL + (1 − pH )yH ) = 0,
yL + σ(iELB − r∗ − σ −1 sL ) − (Mee + σκ)(pL yL + (1 − pL )yH ) = 0,2
VH = −yH+ β(pH VL + (1 − pH )VH ),VL = −yL2 + β(pL VL + (1 − pL )VH ).
The solution to the system of equations is given by the following:yH = 0,iH = r∗ + σ −1 sH + σ −1 (Mee + κσ)pH yL ,−βpH yL2,
VH =(1 − β)(1 − β(pL − pH ))yL = −φ =−iELB + r∗ + σ −1 sL,
σ −1 (1 − (Mee + κσ)pL )iL = iELB ,−yL2 + β(1 − pL )VH
VL =.
(1 − βpL )Solving for the sustainable equilibrium: The Lagrangean is given by
L ≡ E0∞Xβ t −yt2 + 2φt (yt + σ (it − r∗ ) − st − (Mee + σκ)Et yt+1 )t=0+ψt −Et

∞
X
j=0

= E0

∞
X
t=0



2
β j yt+j− WdN (st ) ,




1
t
2
∗
N
β −Ψt yt + 2φt (yt + σ (it − r ) − st ) − 2φt−1 (Mee + σκ)yt − ψt Wd (st ) ,βwhere Ψt = ψ−1 +ψ0 +...+ψt < ∞ is the sum of the Lagrange multipliers on the sustainability
constraint. The FOCs are given by∂yt :−Ψt yt − φt + β −1 (Mee + σκ)φt−1 = 0,∂φt :yt + σ (it − r∗ ) − st − (Mee + σκ)Et yt+1 = 0.

Normalizing the first equation by Ψt , we have−yt − φ̃t + β −1 (Mee + σκ)zt φ̃t−1 = 0,
where φ̃t = φt /Ψt and zt = Ψt−1 /Ψt ∈ (0, 1]. The Karush-Kuhn-Tucker conditions (KKTCs)
must be satisfied as well
φt (it − iELB ) = 0,φt ≥ 0,ψt (Vt − W (st )) = 0,ψt ≥ 0.

34

The initial conditions on the Lagrange multipliers are such that φ−1 = 0 and ψ−1 = 1, which
implies φ̃−1 = 0 and Ψ−1 = 1.

C

Time-iteration method

We explain details of the time-iteration method for the model with a static Phillips curve.
The method is extended to solve other models considered in this paper—the model with astatic Euler equation and the standard New Keynesian model. We explicitly consider a vectorof policy and value functions ς(ξj ) = [y(ξj ), i(ξj ), φ̃0 (ξj ), z(ξj ), V (ξj )]0 as functions of the statevariables ξj = (φ̃, sj ) for j = 1, ..., Ns . We have the following a system of functional equations:e1 (ξj ) ≡ − y(ξj ) − φ0 (ξj ) + β −1 (Mee + σκ)z(ξj )φ̃ = 0,∗e2 (ξj ) ≡ − y(ξj ) − σ (i(ξj ) − r ) + sj + (Mee + σκ)NsXp(sk |sj )y(φ̃0 (ξj ), sk ) = 0,k=1e3 (ξj ) ≡ − V (ξj ) − y(ξj )2 + βNsXp(sk |sj )V (φ̃0 (ξj ), sk ) = 0.

k=1Note that we also have two occasionally binding constraintsi(ξj ) ≥ iELB ,V (ξj ) ≥ WdN (sj ).
Algorithm The time iteration method takes the following steps. For any finite N , WdN (sj )and the policy functions ς(ξj ) are jointly determined with nested loops. For N = ∞, Wd∞ (sj )is the value function under the optimal discretionary policy, and we do not need the steps 5
and 6 below.
0. Make an initial guess for the continuation value under N -period punishment WdN (sj )for j = 1, ..., Ns .
1. Make an initial guess for the policy function ς (0) (ξj ) for j = 1, ..., Ns .
2. For n = 1, 2, ... (n is an index for the number of iteration), given the policy function
previously obtained ς (n−1) (ξj ) for each j, solve− y − φ̃0 + β −1 (Mee + σκ)z φ̃ = 0,
− y − σ (i − r∗ ) + sj + (Mee + σκ)NsXp(sk |sj )y (n−1) (φ̃0 , sk ) = 0,

k=1

− V − y2 + βNsXp(sk |sj )V (n−1) (φ̃0 , sk ) = 0,k=1i ≥ iELB ,V ≥ WdN (sj ),35

for (y, r, φ̃0 , z, V ).
0

3. Update the policy function by setting y = y (n) (ξj ), i = i(n) (ξj ), φ̃0 = φ̃ (n) (ξj ), z =z (n) (ξj ), V = V (n) (ξj ) for j = 1, ..., Ns .
4. Repeat 2-3 until ς (n) (ξj ) converges for each j. That is, ς (n) (ξj ) − ς (n−1) (ξj ) is smallenough.
5. Compute WdN (sj ) recursively by
Wd0 (sj ) = V (0, sj ),WdN (sj ) = maxy,iyd0 (sj ) = y(0, sj ),2−y + βNsX!
p(sk |sj )WdN −1 (sk )

k=1
Ns
X

ydN (sj ) = arg max −y 2 + β
y,i

,
!

p(sk |sj )WdN −1 (sk ) ,k=1

sunject to− y − σ (i − r∗ ) + sj + (Mee + σκ)NsXp(sk |sj )ydN −1 (sk ) = 0,k=1i ≥ iELB .
6. Repeat 1-5 until WdN (sj ) converges for each j.
In Step 2, there are four patterns of binding constraints due to the KKTCs.
(i) i > iELB and V > WdN (sj ): We immediately know φ̃0 = 0 and z = 1. Note that,taking the policy functions in the previous iteration, y (n−1) (φ̃0 , sk ) and V (n−1) (φ̃0 , sk ) as
given, the expected values of the next period’s variables are obtained by evaluating thepolicy functions at φ̃0 = 0. Then we obtainy = β −1 (Mee + σκ)φ̃,i = r∗ + (σ −1 Mee + κ)NsXp(sk |sj )y (n−1) (0, sk ) − σ −1 y + σ −1 sj ,k=1

V = −y 2 + βNsXp(sk |sj )V (n−1) (0, sk ).

k=1(ii) i ≤ iELB and V > WdN (sj ): φ̃0 > 0 and z = 1. The ZLB is binding, i = iELB .

36

(y, φ̃0 ) are obtained by solving the following nonlinear equations:y = −φ̃0 + β −1 (Mee + σκ)φ̃,iELB = r∗ + (σ −1 Mee + κ)NsXp(sk |sj )y (n−1) (φ̃0 , sk ) − σ −1 y + σ −1 sj .

k=1(iii) i > iELB and V ≤ WdN (sj ): φ̃0 = 0 and z ∈ (0, 1). The sustainability constraint isbinding, V = WdN (sj ). Then we obtainy=−W (st ) + β

Ns
X

! 12
p(sk |sj )V (n−1) (0, sk ),k=1

i = r∗ + (σ −1 Mee + κ)NsXp(sk |sj )y (n−1) (0, sk ) − σ −1 y + σ −1 sj ,k=1

z = β(Mee + σκ)

−1

y/φ̃.

(iv) i ≤ iELB and V ≤ WdN (sj ): φ̃0 > 0 and z ∈ (0, 1). Both constraints are binding,i = iELB and V = WdN (sj ). (y, φ̃0 ) are obtained by solving the following nonlinear
equations:
WdN (sk ) = −y 2 + βNsXp(sk |sj )V (n−1) (φ̃0 , sk ),k=1

iELB = r∗ + (σ −1 Mee + κ)NsXp(sk |sj )y (n−1) (φ̃0 , sk ) − σ −1 y + σ −1 sj .

k=1

Note that these equations do not depend on φ̃. Having (y, φ̃0 ) at hand, z = β(Mee +σκ)−1 (y + φ̃0 )/φ̃ is also obtained.
¯
In solving the problem, we discretize the state space of φ̃ ∈ [0, φ̃]. We use Nφ = 401 points¯
and Ns = 2 points respectively for each state variable. The upper bound φ̃ is endogenously
set so that the sequence of the Lagrange multipliers {φ̃t } under consecutive negative shocks
st = sL for t = 1, 2, ..., T1 are bounded above.30 We divide the state space by log-spaced grid
points so that more grid points are toward the origin. We use piecewise-linear functions toapproximate the policy functions off the grid points.
Figure 11 shows the impulse response of the residual functions {e1,t , e2,t , e3,t } under the
OSP with N = 6031 . As in the impulse responses in Figure 1 and 2 in the main text, thecrisis shock hits the economy at time 1 and stays until time 8. We can see that all the valuesof the residual functions are reasonably small. When the sustainability constraint binds attime 9, the value of e1,t increases as zt < 1 and the equation becomes nonlinear. Also, theTaking as given the cutoff probability p̄ = 0.005, T1 is set to the highest integer that satisfies pTL1 > p̄.
We cannot calculate the Euler equation errors under the OSP with N = 20 as the only analytical solutions
are available.
30

31

37

values of e3,t increase when the values of Vt are in transition from one value to another (i.e.,
in each of the first few periods after time 1 and after time 9).
Figure 11: Residuals (under the OSP with N = 60)
10 -3

1

|e 1 |

10 -3

1|e 2 |

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

1

10

20

Value

-20

0

1

10

0

20

30

1

20

0.8

10

0.6

|e 3 |

10 -3

1

1

10

20

z

-40
-60
-80
-100

1

10

20

0

0.4
1

10

20

1

10

20

Table 5 shows the the average and maximum of the residuals in absolute terms (log 10
units) based on a 100,000-period stochastic simulation under the OCP, the OSP with N = ∞and the OSP with N = 60. The values of e3 tend to be worse than others as we use linear
interpolation, whereas the value functions are at least quadratic. Note that the equation fore1 holds with equality (up to the machine precision) under the OCP, as the equation is linear(i.e., z = 1) and we use the equation to substitute variables other than the ones we solve forwith the other equations.
Table 5: Residuals
|e1 |
OCP
OSP
with N = ∞with N = 60

|e2 |

|e3 |

L1
-16.59L∞-14.15

L1
-7.55

L∞
-3.48

L1
-4.83

L∞
-2.36

-7.43
-6.22

-3.38
-3.46

-7.27
-8.01

-3.66
-3.94

-4.84
-4.99

-2.28
-2.66

Notes: L1 is the average and L∞ is the maximum of the residuals in absolute terms (log 10
units) based on a 100,000-period stochastic simulation.
38

D

A detailed account of the model with a static Phillips curve

In this section, we provide a detailed account of the dynamics of the model with a static
Phillips curve from the vantage point of the policy functions. There are two purposes. First,
seeing the dynamics of the economy in this way helps us better understand what’s happening
in the model. Second, it helps us to understand why the dynamics of the economy are history
independent when N is sufficiently small and why it is possible to solve the model in this casein an alternative way that we later describe in Appendix E.
We can accomplish the first with the baseline calibration of the main text, but cannot
accomplish the second because the time-iteration does not converge for any small values of
N consistent with history independent dynamics. Accordingly, we will use an alternative
calibration in which the time-iteration method converges for some values of N consistent withhistory independent dynamics. In this alternative calibration, pH = 0.2/100 and pL = 0.5.
The values for other parameters are the same as the baseline values from the main text.
We first describe the dynamics of the economy under the optimal commitment policy(OCP). We then describe the dynamics of the economy under optimal sustainable policies(OSPs) with N = ∞ and N = 120.

D.1

Optimal commitment policy

Suppose that the economy is initially at its risky steady state at time 0 where the policy
rate is positive and the Lagrange multiplier is zero. The economy falls into the crisis state attime 1 and stays there until time 4. The economy is back in the normal state from time 5 on.
Figure 12: Dynamics with an alternative calibration
3

Policy rate (ann. %)Output gap (%)
2

2

Inflation (ann. %)

1

4

0.5

0
0

-2
1

OCP

-4

-0.5

OSP w/ N=
OSP w/ N=120

-6
0

0

5

10

ODP

0

5

10

-1

0

5

10

Note: ODP, OCP, and OSP stand for optimal discretionary policy, optimal commitment policy, and optimal
sustainable policy, respectively. The policy rate and the inflation rate are expressed in annualized percent.
The output gap is expressed in percent.

The economy’s dynamics under the OCP in this recession scenario are shown by the solid
grey lines in Figure 12. Consistent with what we saw in the main body of the paper, thecentral bank keeps the policy rate at the ELB even after the crisis shock disappears andengineers an overheating of the economy.
The solid black lines in Figure 13 are the value and policy functions associated with the
OCP. The black dots in Figure 13 trace the dynamics of the economy in this recession scenario
along the value and policy functions. The dynamics of the economy from time 1 to time 4
are governed by the crisis-state value and policy functions shown in the right panel. As the
economy stays in the crisis state, the Lagrange multiplier—the sole endogenous state variable39

of the model—increases. The pace of the increase decelerates the longer the economy is in thecrisis state. If the economy were to stay in the crisis state forever, the Lagrange multiplierconverges to a finite value. The evolution of output in the crisis state mirrors that of the
Lagrange multiplier. As the Lagrange multiplier increases, output increases. If the economy
were to stay in the crisis state forever (which is a zero probability event), output converges
to a finite value, just as the Lagrange multiplier converges to a finite value.
Figure 13: Value/policy functions—OCP and ODP—
s=s h5

Output

OCP
ODP

-5
0
0

Policy rate

s=s l

0

5

10

15

0

5

10

15

0

5

10

15

0
0

5

10

15

5

10

15

0

0

-0.5

-0.5

-1

0

5

10

-1

15

Lagrange
multiplier

30
10

20

5

10

0
0

5

10

15

Value

0

-50

-50
-100
-100

0

5

10

15

0

-1

-1

Note: ODP and OCP stand for optimal discretionary policy and optimal commitment policy, respectively.
The policy rate and the inflation rate are expressed in annualized percent. The output gap is expressed inpercent.

Once the economy is back in the normal state, the dynamics of the economy are governedby the normal-state value and policy functions shown in the left panel. In the first period
back in the normal state, the lagged Lagrange multiplier is positive, which implies positive
output at time 1. As the time continues, the Lagrange multiplier gradually declines, eventually
returns to zero. Together with the Lagrange multiplier, output declines as the normal statecontinues, eventually returning to zero.

D.2

Optimal sustainable policies

The economy’s dynamics under the OSPs with N = ∞ and N = 120 in this recession
scenario are shown by the solid and dash-dotted black lines in Figure 12. For the set ofparameter values considered in this section, the sustainability constraint binds right after thecrisis state ends even under the OSP with N = ∞, shortening the ELB duration and limiting40

the size of output and inflation overshoots. With N = 120, the ELB duration is shorter andthe size of the overshoot is smaller than with N = ∞.
The value/policy functions associated with the OSPs with N = ∞ and N = 120 are shownby the solid black lines in Figure 14 and 15, respectively. The dynamics of the economy under
the OSPs in this recession scenario are traced by the black dots in these figures.
Figure 14: Value/policy functions—OSP with N = ∞—s=s h
5

OutputOCP
OSP w/ N=
ODP

-5
0
0

Policy rate

s=s l

0

5

10

15

0

5

10

15

0

5

10

15

0
0

5

10

15

5

10

15

0

0

-0.5

-0.5

-1

0

5

10

-1

15

Lagrange
multiplier

30
10

20

5

10

0
0

5

10

15

Value

0

-50

-50
-100
-100

0

5

10

15

0

-1

-1

Note: ODP, OCP, and OSP stand for optimal discretionary policy, optimal commitment policy, and optimal
sustainable policy, respectively. The policy rate and the inflation rate are expressed in annualized percent.
The output gap is expressed in percent.

Under the OSP with N = ∞, the crisis-state Lagrange multiplier increases as the crisis
state persists, as seen in the right column of Figure 14. The crisis-state output increases
with the Lagrange multiplier, but only when the lagged Lagrange multiplier is very small.
Otherwise, the crisis-state policy function for output is flat. Once the economy returns to
the normal state, the dynamics of the economy are governed by the normal-state value andpolicy functions shown in the left column of Figure 14. The sustainability constraint binds if
the lagged Lagrange multiplier is sufficiently large. For those values of the lagged Lagrange
multiplier, the value and policy functions are flat. As a result, the Lagrange multiplier takes
a low positive value in the first period of the normal state. Thus, the Lagrange multiplierdeclines quickly under the OSP with N = ∞ than under the OCP.
Under the OSP with N = 120, the crisis-state Lagrange multiplier increases as the crisis
state persists, as seen in the right column of Figure 15. However, the policy function for
output is flat so that output is constant in the crisis state. When the economy returns to thenormal state, the dynamics of the economy are governed by the normal-state value and policy
41

Figure 15: Value/policy functions—OSP with N = 120—s=s h
5OutputOCP
OSP w/ N=120
ODP

-5
0
0

Policy rate

s=s l

0

5

10

15

0

5

10

15

0

5

10

15

0
0

5

10

15

5

10

15

0

0

-0.5

-0.5

-1

0

5

10

-1

15

Lagrange
multiplier

30
10

20

5

10

0
0

5

10

15

Value

0

-50

-50
-100
-100

0

5

10

15

0

-1

-1

Note: ODP, OCP, and OSP stand for optimal discretionary policy, optimal commitment policy, and optimal
sustainable policy, respectively. The policy rate and the inflation rate are expressed in annualized percent.
The output gap is expressed in percent.

functions shown in the left column of Figure 14. The normal-state value and policy functions
are qualitatively similar to those under the OSP with N = ∞. However, one interesting
feature of the economy under the OSP with N = 120 is that the value of the lagged Lagrange
multiplier in the first period of the normal state does not depend on the realized crisis shock
duration. No matter how long the crisis shock lasts, the Lagrange multiplier takes a certain
value given by the flat part of the normal-state policy function for the Lagrange multiplier.
As a result, post-crisis dynamics do not depend on the realized crisis shock duration. In otherwords, the OSP with N = 120 is history independent. In this case, the dynamics of theeconomy in the normal state can be fully characterized by a finite sequence of output, policyrate, Lagrange multiplier, and the value.
Note that the key difference between the history dependent and history independent cases
is whether the value of the Lagrange multiplier in the first period of the crisis state is above
or below the value of the lagged Lagrange multiplier above which the sustainability constraint
binds in the normal state. This threshold value of the lagged Lagrange multiplier is indicatedby thin vertical lines in the left panels of Figure 14 and 15.
If the value of the Lagrange multiplier in the first period of the crisis state is above thethreshold value of the lagged Lagrange multiplier, then the value of the Lagrange multiplier inany period of the crisis state is also above the threshold value of the lagged Lagrange multiplierbecause the Lagrange multiplier increases as the crisis persists. In this case, when the economy
42returns to the normal state, output, inflation, value, policy rate, and the Lagrange multiplierin the first period of the normal state will be given by the value of the normal state policy
function in the flat region where the sustainability constraint binds, regardless of how long thecrisis lasts. Thus, the allocations in the first period back in the normal state do no depend on
the realized crisis shock duration. If the Lagrange multiplier in the first period of the normal
state does not the realized crisis duration, the dynamics of the economy will not depend on
the realized crisis shock duration.
If the value of the Lagrange multiplier in the first period of the crisis state is below thethreshold value of the lagged Lagrange multiplier, then the allocations in the first period backin the normal state depend will differ depending on whether the realized crisis duration is
one period or longer. This is because the normal-state value and policy functions are not in
the flat region. In this case, the economy’s dynamics exhibit history dependence. The higher
the threshold value of the lagged Lagrange multiplier is relative to the value of the Lagrange
multiplier in the first period of the crisis state, the more history dependent the economy’s
dynamics become.

E

Two alternative solution methods when N is low

The previous section describes why the dynamics are history independent when N is
sufficiently small and suggests that the dynamics of the economy are fully characterized by avector of scalars satisfying a system of certain nonlinear equations and inequality constraints.
As a result, it is not necessary to rely on the time-iteration method to compute the dynamics
of the model. In this section, we provide the details of the solution algorithm we use whenthe economy’s dynamics are history independent.
We also show that even if the dynamics are history dependent for some low N , we can exploit the fact that the policy functions are flat above the threshold of the Lagrange multiplier.
This complements the standard numerical method with time iteration explained in Section C
and the analytical method to be explained below. As a result, we can solve the model for all
N ∈ R+ by using either of the solution methods. We confirmed that these solution methods
yield the same allocations when multiple solution methods are available for the same N .
Analytical method
The big picture of the solution algorithm is as follows. Assuming that the additional
ELB duration is τ , compute the allocation satisfying the model’s equilibrium conditions thatare captured by equality constraints. Check (i) whether the model’s inequality constraints(ELB and sustainability constraints) are satisfied and (ii) whether the history independence
assumption is indeed valid. If these two requirements are met, then we have found the solution.
If not, continue to search for the value of τ that satisfies all the equilibrium conditions, boththose represented by equality and inequality constraints.
As discussed in Section 2, this alternative solution method works in the model with astatic Phillips curve, but not in the model with a forward-looking Phillips curve. As analyzed
in Nakata and Schmidt (2019a), in the model with a forward-looking Phillips curve, when thecrisis shock is recurring, inflation and the output gap in the normal state are not zero evenafter the effect of a past crisis dissipated. However, in the model with a static Phillips curve,
43

inflation and the output gap converge to zero in the normal state once the effect of a past crisis
shock dissipates (because there is no trade-off between inflation and output stabilization). The
alternative solution method we describe below takes advantage of this feature of the modelwith a static Phillips curve.32
Below, we first describe the set of equality and inequality constraints the solution has tosatisfy given τ . We then describe how to verify that the history independence assumption issatisfied.
Case 1: τ = 0 (liftoff occurs immediately after the crisis shock disappears)
We describe how to find the solution to the model if the economy’s dynamics are history independent and the policy rate is above the ELB right after the crisis shock disappears. In thiscase, the dynamics of the economy are fully characterized by {iH , iM , iL , yH , yM , yL , vH , vM , vL }where the subscript denote one of the three states of this economy:• In L state, the crisis shock is present. Only the ELB constraint is binding.
• In M state, the crisis shock is absent. Only the sustainability constraint is binding.
This state follows L state.
• In H state, the crisis shock is absent. No constraint is binding.
By definition of this case, the following equality and inequality constraints must be satisfied:
iH > −r∗iM > −r∗iL = −r∗andvH > vP unish,H (vH , vL , yH , yL )vM = vP unish,H (vH , vL , yH , yL )vL > vP unish,L (vH , vL , yH , yL )where vP unish,H (vH , vL , yH , yL ) and vP unish,L (vH , vL , yH , yL ) are given by:


i−1
i
i
vD,H
= β (1 − pH )vD,H+ pH vD,L
,


i−1
i−1 2
i
i
vD,L = −(yD,L ) + β (1 − pL )vD,H + pL vD,L
,


i−1
i
i
yD,L
= − σ (−r∗ − sL ) + (Mee + σκ) (1 − pL )yD,H+ pL yD,L,
K , v K , y K , y K ) = (v , v , y , y ) and v
0for i = 1, ..., K, where (vD,H
H L H LP unish,H = vD,H . K is
D,L D,H D,L

32

Note that we could also use this alternative solution method to the model with perfectly sticky prices—in
which the inflation rate is zero at any time and after any history of shocks—. In the model with perfectly
sticky prices, there is no inflation-output trade-off and the output gap eventually converges to zero once theeffect of a past crisis dissipates.

44

the length of punishment. In this case, the equilibrium conditions are given by:
yH = (Mee + σκ) [(1 − pH )yH + pH yL ] − σ (iH − sH ) ,yM = (Mee + σκ) [(1 − pH )yH + pH yL ] − σ (iM − sH ) ,yL = (Mee + σκ) [(1 − pL )yM + pL yL ] − σ (−r∗ − sL ) ,2
vH = −yH+ β [(1 − pH )vH + pH vL ] ,2vM = −yM+ β [(1 − pH )vH + pH vL ] = vP unish,H (vH , vL , yH , yL ),vL = −yL2 + β [(1 − pL )vM + pL vL ] ,where we used iL = −r∗ . Because the central bank only cares about output, yH = 0. Thus,
iH = σ −1 (Mee + σκ)pH yL + sH ,iM = σ −1 (Mee + σκ)pH yL + sH − σyM ,yL = (Mee + σκ) [(1 − pL )yM + pL yL ] − σ (−r∗ − sL ) ,vH = β [(1 − pH )vH + pH vL ] ,2vM = −yM+ β [(1 − pH )vH + pH vL ] ,vL = −yL2 + β [(1 − pL )vM + pL vL ] ,vM = vP unish,H (vH , vL , yH , yL )There are 7 equations and 7 unknowns: (iH , iM , yM , yL , vH , vM , vL ).
Guess (yL , vL ). Then,• From the first three equations, we can determine (iH , iM , yM ).
• From the fifth and sixth equations, we can determine (vH , vM ).
We need to check whether the fourth and seventh equations hold:vH = β [(1 − pH )vH + pH vL ] ,vM = vP unish,H (vH , vL , yH , yL )That is, we can basically reduce the system to two-unknowns in two equations. Once you
solve the system of equations, we need to verify the following four inequalities:
iH > −r∗iM > −r∗vH > vP unish,H (vH , vL , yH , yL )vL > vP unish,L (vH , vL , yH , yL )When τ = 0, the Lagrange multiplier is zero in the first period after the crisis shock
disappears. Thus, the dynamics of the economy cannot depend on the realized duration of
the shock. That is, the economy’s dynamics are history independent by construction.
Case 2: τ > 0 (liftoff occurs at least two periods after the crisis shock disappears)

45

In this section, we discuss how to solve for cases in which the policy rate stays at the ZLB
for at least one period after the crisis shock disappears:• In L state, the crisis shock hits the economy. Only the ZLB constraint is binding.
• In M0 state, the crisis shock is absent. Both the ZLB and sustainability constraints arebinding. This state follows L state.
• In Mi state from i = 1 to i = τ − 1, the crisis shock is absent. Only the ZLB constraint
is binding. This state follows Mi−1 state.
• In Mτ state, the crisis shock is absent. No constraint is binding. This state follows
Mτ −1 state.
• In H state, the crisis shock is absent. No constraint is binding. And, output gap is zero.
By construction, the following equality and inequality constraints must be satisfied:
iH > −r∗iM,τ > −r∗iM,τ −1 = −r∗...
iM,1 = −r∗iM,0 = −r∗iL = −r∗andvH > vP unish,H (vH , vL , yH , yL )vM,τ > vP unish,H (vH , vL , yH , yL )vM,τ −1 > vP unish,H (vH , vL , yH , yL )...
vM,1 > vP unish,H (vH , vL , yH , yL )vM,0 = vP unish,H (vH , vL , yH , yL )vL > vP unish,L (vH , vL , yH , yL )where vP unish,H (vH , vL , yH , yL ) and vP unish,L (vH , vL , yH , yL ) are given by:


i−1
i
i
vD,H
= β (1 − pH )vD,H+ pH vD,L
,


i−1
i−1 2
i
i
vD,L = −(yD,L ) + β (1 − pL )vD,H + pL vD,L
,


i−1
i
i
yD,L
= − σ (−r∗ − sL ) + (Mee + σκ) (1 − pL )yD,H+ pL yD,L,0
K , v K , y K , y K ) = (v , v , y , y ) and v
for i = 1, ..., K, where (vD,H
H L H LP unish,H = vD,H . K is
D,L D,H D,L
the length of punishment. In this case, the equilibrium conditions are given by:46

yH = (Mee + σκ) [(1 − pH )yH + pH yL ] − σ (iH − sH ) ,yM,τ = (Mee + σκ) [(1 − pH )yH + pH yL ] − σ (iM,τ − sH ) ,yM,τ −1 = (Mee + σκ) [(1 − pH )yM,τ + pH yL ] − σ (−r∗ − sH ) ,...
yM,k = (Mee + σκ) [(1 − pH )yM,k+1 + pH yL ] − σ (−r∗ − sH ) ,...
yM,1 = (Mee + σκ) [(1 − pH )yM,2 + pH yL ] − σ (−r∗ − sH ) ,yM,0 = (Mee + σκ) [(1 − pH )yM,1 + pH yL ] − σ (−r∗ − sH ) ,yL = (Mee + σκ) [(1 − pL )yM,0 + pL yL ] − σ (−r∗ − sL ) ,and2vH = −yH+ β [(1 − pH )vH + pH vL ] ,2vM,τ = −yM,N+ β [(1 − pH )vH + pH vL ] ,2vM,τ −1 = −yM,τ
−1 + β [(1 − pH )vM,τ + pH vL ] ,...
2
vM,k = −yM,k+ β [(1 − pH )vM,k+1 + pH vL ] ,...
2
vM,1 = −yM,1+ β [(1 − pH )vM,2 + pH vL ] ,2
vM,0 = −yM,0+ β [(1 − pH )vM,1 + pH vL ] ,vL = −yL2 + β [(1 − pL )vM,0 + pL vL ] ,vM,1 = vP unish,H (vH , vL , yH , yL )Because the central bank only cares about output stabilization, yH = 0. Using yH = 0, we
obtainiH = σ −1 (Mee + σκ)pH yL + sH ,iM,τ = σ −1 (Mee + σκ)pH yL + sH − σ −1 yM,τ ,yM,τ −1 = (Mee + σκ) [(1 − pL )yM,τ + pL yL ] − σ (−r∗ − sH ) ,...
yM,k = (Mee + σκ) [(1 − pL )yM,k+1 + pL yL ] − σ −1 (−r∗ − sH ) ,...
yM,1 = (Mee + σκ) [(1 − pL )yM,2 + pL yL ] − σ (−r∗ − sH ) ,yM,0 = (Mee + σκ) [(1 − pL )yM,1 + pL yL ] − σ (−r∗ − sH ) ,yL = (Mee + σκ) [(1 − pL )yM,0 + pL yL ] − σ (−r∗ − sL ) ,

47

and
vH = β [(1 − pH )vH + pH vL ] ,2vM,τ = −yM,τ
+ β [(1 − pH )vH + pH vL ] ,2vM,τ −1 = −yM,τ
−1 + β [(1 − pH )vM,τ + pH vL ] ,...
2
vM,k = −yM,k+ β [(1 − pH )vM,k+1 + pH vL ] ,...
2
vM,1 = −yM,1+ β [(1 − pH )vM,2 + pH vL ] ,2
vM,0 = −yM,0+ β [(1 − pH )vM,1 + pH vL ] ,vL = −yL2 + β [(1 − pL )vM,0 + pL vL ] ,vM,1 = vP unish,H (vH , vL , yH , yL )We are solving for (yL , iH , iM,τ ), {yM,i }τi=0 , (vH , vL ) and {vM,i }τi=0 . 2(τ + 1) + 5 unknowns
in 2(τ + 1) + 5 equations.
Guess (yL , vL ). Then,• From the first τ + 3 equations, we can determine (iH , iM,τ ) and {yM,i }τi=0 .
• From the equations for vM,i and vL (τ +2 equations), we can determine vH and {yM,i }τi=0 .
Then, we need to check whether the following two equations hold:vH = β [(1 − pH )vH + pH vL ] ,vM = vP unish,H (vH , vL , yH , yL )That is, we can basically reduce the system to two unknowns in two equations. Once you
solve the system of equations, we need to verify the following τ + 4 inequalities:
iH > −r∗iM,τ > −r∗vH > vP unish,H (vH , vL , yH , yL )vM,τ > vP unish,H (vH , vL , yH , yL )...
vM,1 > vP unish,H (vH , vL , yH , yL )vL > vP unish,L (vH , vL , yH , yL )If these inequality constraints are satisfied, then we have the right τ and proceed to verify thehistory independence assumption. Otherwise, we adjust τ and solve the system of equations
described above again.
Alternative numerical method
Next, we modify the time-iteration method to solve the model numerically for some low
N s when the solution is not history-independent. We use the fact that the policy functions
48

are flat above the threshold of the Lagrange multiplier φ̃ (which is the only endogenous statevariable in the model) for each state. Roughly speaking, the alternative numerical method
works as follows. First, taking the policy functions as given, we solve for numbers of theoutput and value (ȳH , v̄H , ȳL , v̄L ) and the threshold (φ̃∗H , φ̃∗L ) when the policy function isflat and the current period’s state variable φ̃ is irrelevant for each state. Then, we use theequilibrium conditions to update the policy functions only for φ̃ < φ̃∗ . We iterate this processuntil the policy functions converge.
Followings are the details. First of all, we use the fact that the following equations holdin the low state s = sL for a sufficiently high Lagrange multiplier φ̃:ȳL = (Mee + κσ) (pL ȳL + (1 − pL )ȳH ) + σ(r∗ + sL ),V̄L = −ȳL2 + βpL V̄L + β(1 − pL )V̄H .
That is, the output and value yL = ȳL and VL = V̄L are constant in the current period
regardless of the value of φ̃ (as shown in Figure 14 and 15). This is the case when the next
period is in the high state s = sH and the sustainability constraint will be binding. Then,the next period’s output and value in the high state are also constant, and given byȳH = (Mee + κσ) pH y(φ̃∗∗ , sL ) + (1 − pH )y(φ̃∗∗ , sH ) + σ(r∗ + sH ),2
V̄H = W (sH ) = −ȳH+ βpH V (φ̃∗∗ , sL ) + β(1 − pH )V (φ̃∗∗ , sH ),where φ̃0 = φ̃∗∗ is the value of the (two periods ahead) Lagrange multiplier after the sustainability constraint is binding in the next period, which is also known to be constant. See theflat region of the policy function for the Lagrange multiplier in the left panel of Figure 14
and 15. These equations can be solved for (ȳH , V̄H , φ̃∗∗ ) taking the policy functions y(φ, s)and V (φ, s) as given. Once we have (ȳH , V̄H ), we can also compute (ȳL , V̄L ).
Then, let φ̃∗H the threshold of the Lagrange multiplier above which the output and value
in the high state is constant. Taking the policy function φ̃0 (φ̃, s) as given, such a thresholdmust satisfy φ̃∗∗ = φ̃0 (φ̃∗H , sH ). Also, let φ̃∗L be the threshold of the Lagrange multiplier above
which the output and value in the low state are constant. As the ELB always binds in the
low state, we have the following equation from the first-order condition
φ̃∗H = −ȳL + β −1 (Mee + κσ)φ̃∗L .
Therefore we obtain φ̃∗L =β(φ̃∗H +ȳL )Mee +κσ .

Then we use the equilibrium conditions to update thepolicy functions at each grid point of φ̃, but we do so only for φ̃ < φ̃∗H in the high state andfor φ̃ < φ̃∗L in the low state. The ELB may or may not bind in the high state (Case (i) or (ii)in Appendix C) whereas the ELB always binds in the low state (Case (ii) in Appendix C).
We also know that the sustainability constraint is always slack, which is greatly helpful to geta convergent result. When φ̃ is above the threshold, we set y = ȳH , φ̃0 = φ̃∗∗ , and V = V̄H
49

for φ̃ ≥ φ̃∗H in the high state, and y = ȳL , φ̃0 = −y¯L + β −1 (1 + κσ)φ̃, and V = V̄L for φ̃ ≥ φ̃∗Lin the low state.33 Once we update the policy functions y(φ̃, s), φ̃0 (φ̃, s), and V (φ̃, s), we also
renew the value of reneging WdN (s) recursively.
With the parameter values we use except for N , the altenative numerical solutions areavailable for 12 ≤ N ≤ 59 whereas the original numerical solutions are computed for N ≥ 39.
Below N < 12, only the history-independent analytical solutions are available. Note that theanalytical solutions are valid only when N is low, as the assumption of history independenceis violated otherwise. The gap between the analytical solution and the numerical soluiton is
at most 0.002% for N ≤ 24. Also, the gap between two numerical solutions is at most 0.001%
for 39 ≤ N ≤ 59.

F

Results in the model with a static Phillips curve and discounted Euler equation
Figure 16 shows the dynamics of the economy under the ODP, the OCP, and OSPs with

N = [20, 60, ∞] and discounted Euler equations for the cases with Mee = [0.8, 0.5, 0.2]. As
in the baseline case with Mee = 1.0, the responses of the policy rate, the output gap, andinflation under OSPs are in between those under the ODP and those under the OCP. As Meegets lower, the commitment effect is diminished and the degree of the overshooting under the
OCP is smaller. Under OSPs, the overshoot is even smaller with lower N s.34
Also, OSPs show history independence. The highest value of N under which the OSP
exhibits history independence is increasing as the future output gap is more discounted.

33
Note that φ̃∗L can be below zero. If so, yL = ȳL regardless of φ̃ in the low state and the solution ishistory-independent.
34
As in the baseline case, the standard time iteration method does not converge with some low N s. We use
the alternative numerical method (when the solution is history-dependent) or the analytical method (whenthe solution is history-independent) to solve for the allocations.

50

Figure 16: Dynamics—Model with discounted Euler equations—(i) Mee = 0.8
Policy Rate (ann. %)3

Output Gap (%)

4

Inflation (ann. %)

0.5

2
2

0

0
-2

1

-0.5

-4

OCP/OSP w/ N=
OSP w/ N=60
OSP w/ N=20

-6
0

1

10

20

ODP

1

10

20

-1

1

10

20

(ii) Mee = 0.5
3

Policy Rate (ann. %)Output Gap (%)

2

Inflation (ann. %)
0

0
2

-0.2

-2

-0.4
-4

1

-0.6

OCP/OSP w/ N=
OSP w/ N=60

-6

-0.8

-8

-1

OSP w/ N=20
ODP

0

1

10

20

1

10

20

1

10

20

(iii) Mee = 0.2
3

Policy Rate (ann. %)Output Gap (%)

2

Inflation (ann. %)
0

0
-0.2

2

-2

-0.4

-4

1

-0.6

-6

-0.8

-8

-1

OCP/OSP w/ N=
OSP w/ N=60
OSP w/ N=20
ODP

0

1

10

20

1

10

20

1

10

20

Notes: Parameter values are β = 0.9925, σ = 1.0, κ = 0.25/7, pH = 0.005, and pL = 0.75. sL
is chosen so that yL = −7.0.

51

GModel with a static Euler equation
The policymaker maximizes∞
X

V0 = −E0
β t πt2 + λyt2 ,t=0

subject to(18)

πt = κyt + β̃Et πt+1 ,
−1

−1

−1

∗

−1

πt − β̃ (κ + σ β̃)yt ≥ −r − σ st ,
∞
X

2
2
Vt = −Et
β j πt+j+ λyt+j≥ WdN (st ),r̃t = β̃

(19)
(20)

j=0

for all t ≥ 0. Mpc ≤ 1.0 is a parameter whose value is less than one to diminish the degreeof forward-lookingness. Note that the right-hand side of Equation (7) is obtained by thestatic version of the Euler equation, −σ −1 yt = r̃t − Et πt+1 and Equation (6), where we definer̃t ≡ it − σ −1 st and β̃ ≡ βMpc . The shock, st , follows two-state Markov chain, st ∈ {sH , sL }1 − pH pH
where sH > sL . Transition probability matrix is given as P =, where pH is
1 − pL pL
the frequency of the crisis and pL is the persistence of the crisis. WdN (st ) is the continuation
value under N -period punishment.
The analytical solution for the discretionary outcome: The ZLB is slack in thehigh/normal state, φH = 0, and the ZLB is binding in the low/crisis state, φL = φ > 0
and r̃L = −r∗ − sL . Thus, the equilibrium conditions become
λyH + κπH = 0,
πH = κyH + β̃ ((1 − pH )πH + pH πL ) ,πH = (κ + σ −1 β̃)yH + βr̃H ,λyL + κπL + σ −1 β̃φ = 0,πL = κyL + β̃ ((1 − pL )πH + pL πL ) ,πL = (κ + σ −1 β̃)yL + βr̃L .
These equations can be solved for (yH , πH , r̃H , yL , πL , φ).

52

Solving for the sustainable equilibrium: The Lagrangean is given by
L ≡ E0∞Xβ t −πt2 − λyt2

t=0





+2φ1t −κyt + πt − β̃Et πt+1 + 2φ2t πt − (κ + σ −1 β̃)yt


∞

X

2
2
+ψt −Et
β j πt+j+ λyt+j− WdN (st ) ,

j=0

= E0

∞
X

n


o

β t −Ψt πt2 + λyt2 + 2φ1t (−κyt + πt ) − 2Mpc φt−1 πt + 2φ2t πt − (κ + σ −1 β̃)yt − ψt WdN (st ) ,t=0

where Ψt = ψ−1 +ψ0 +...+ψt < ∞ is the sum of the Lagrange multipliers on the sustainability
constraint. The FOCs are given by∂yt :−λΨt yt − κφ1t − (κ + σ −1 β̃)φ2t = 0,∂πt :−Ψt πt + φ1t − Mpc φ1t−1 + φ2t = 0.

Normalizing the equations by Ψt , we have− λyt − κφ̃1t − (κ + σ −1 β̃)φ̃2t = 0,
− πt + φ̃1t − zt Mpc φ̃1t−1 + φ̃2t = 0,
where φ̃1t = φ1t /Ψt , φ̃2t = φ2t /Ψt , and zt = Ψt−1 /Ψt ∈ (0, 1]. The Kraush-Kuhn-Tucker
conditions (KKTCs) must be satisfied as well

φ2t (it − iELB ) = 0,
φ2t ≥ 0,ψt Vt − WdN (st ) = 0,ψt ≥ 0.
The initial conditions on the Lagrange multipliers are such that φ1,−1 = φ2,−1 = 0 andψ−1 = 1, which implies φ̃1,−1 = φ̃2,−1 = 0 and Ψ−1 = 1.
Figure 17 shows the dynamic of the economy under the OCP, the ODP, and OSPs with
N = [60, 120, ∞]. The dynamics of the economy under the OCP and the ODP are consistent
with those in the model with a static Phillips curve. However, we can see more differnce
in inflation and the policy rate, whereas the output gap difference is small. The lift off isimmediately after the crisis shock is gone even under the OCP, although the steady-statepolicy rate is lower under the OCP than under the ODP. This makes inflation higher in thenormal state, which also mitigates the inflation drop in the crisis state through expectation
under the OCP.
Under the OSPs (except for N = ∞), the policy maker wants to raise the policy rate53

more than under the OCP but less than under the ODP. This makes inflation lower under
the OSPs than under the OCP. The lower N gets, the lower inflation is under the OSPs.
Figure 17: Dynamics—Model with a static Euler equation—

3

Policy Rate (ann. %)Output Gap (%)

2

Inflation (ann. %)

0.5

0
2

0

-2
-4

1

-0.5

OCP/OSP w/ N=
OSP w/ N=120-6

OSP w/ N=60
ODP

0

1

10

20

-8

1

10

20

-1

1

10

20

Notes: Parameter values are β = 0.9925, σ = 1.0, κ = 0.057, λ = 0.0625, sL = −7.5443,
pH = 0.005, and pL = 0.75. (κ, sL ) is chosen so that yL = −7.0 and πL = −1.0/4.

H

Tenure duration of leadership at central banks
Table 6: Average Tenure Duration of Chairpersons in Select Central Banks
Year ofNo. of leaders

No. of leaders

Average tenure

Average tenure

foundation

since foundation

since 1946

since foundation

since 1946

Federal Reserve System

1914

16

10

6.9

8.1

European Central Bank

1998

3

3

6.5

6.5

Bank of Canada

1934

9

8

9.9

9.7

Bank of Japan

1882

31

15

4.4

4.9

Bank of England

1694

120

9

2.7

8.5

Central BankI

Sveriges Riksbank

1901

14

11

8.1

6.1

Swiss National Bank

1907

14

10

8.1

7.4

Time-inconsistency of the commitment policy in the words ofpolicymakers

The time-inconsistency of the commitment policy at the ELB is not a mere theoretical curiosity. Policymakers in many central banks have pointed out the potential timeinconsistency of the commitment-type forward guidance policy. Some have argued that the54Table 7: Maximum Tenure Duration of Chairpersons in Select Central Banks
Central Bank
Federal Reserve System
European Central Bank

Max duration

Max duration

since foundation

since 1946

18 yrs and 10 months (Martin)

18 yrs and 10 months (Martin)*

8 (Trichet)

8 (Trichet)

Bank of Canada

20 yrs and 4 months (Towers)

14 (Boey)

Bank of Japan

8 yrs and 6 months (Ichimada)

8 yrs and 6 months (Ichimada)Bank of England

24 (Norman)

12 (Cobbold)

Sveriges Riksbank

19 (Rooth)

Swiss National Bank
14 (Bachmann)Note: The tenure of Alan Greenspan lasted for 18 years and 6 months.

18 (Asbrink)
11 (Leutwiler)

time-inconsistency is one key reason for why most central banks refrained from making theoverheating commitment. Below are some examples:

I.1

Bean (2013)“In particular, we signalled our intention not to countenance tightening policy until unemployment has fallen to at least 7 percent.”“This guidance is intended primarily to clarify our reaction function and thus makepolicy more effective, rather than to inject additional stimulus by pre-committing
to a time-inconsistent ‘lower for longer’ policy path in the manner of Woodford(2012). While such a time-inconsistent policy may be desirable in theory, in an
individualistic committee like ours, with a regular turnover of members, it is notpossible to implement a mechanism that would credibly bind future members in
the manner required.”

I.2

Bullard (2013)“The New Keynesian, sticky price literature has been influential in U.S. monetary policymaking. The literature has been led by Michael Woodford. This line of research argues thatpolicy accommodation can be provided even when the policy rate is near zero. The extra
accommodation comes from a promise to maintain the near zero policy rate into the future,beyond the point when ordinary policymaker behavior would call for an increase in the policy
rate. This promise must be credible to have an impact.
The “Woodford period” approach to forward guidance relies on a credible announcement made today that future monetary policy will deviate from normal.
The central bank does not actually behave differently today. One might argue
that such an announcement is unlikely to be believed. Why should future monetary policy deviate from normal once the economy is growing and inflation isrising? But if the announcement is not credible, then the private sector will notreact with more consumption and investment today. That is, any effects would
be minimal.”
55

I.3

Carney (2012)“Today, to achieve a better path for the economy over time, a central bank may need to commit
credibly to maintaining highly accommodative policy even after the economy and, potentially,
inflation picks up. Market participants may doubt the willingness of an inflationtargeting central bank to respect this commitment if inflation goes temporarily
above target. These doubts reduce the effective stimulus of the commitment anddelay the recovery.”

I.4

Clarida (2019)“The benefits of the makeup strategies rest heavily on households and firms believing in advance that the makeup will, in fact, be delivered when the time comes–for example, thata persistent inflation shortfall will be met by future inflation above 2 percent. As is wellknown from the research literature, makeup strategies, in general, are not time
consistent because when the time comes to push inflation above 2 percent, conditions at that time will not warrant doing so. Because of this time inconsistency,
any makeup strategy, to be successful, would have to be understood by the public
to represent a credible commitment. That important real-world consideration isoften neglected in the academic literature, in which central bank “commitment
devices” are simply assumed to exist and be instantly credible on decree. Thus,
one of the most challenging questions is whether the Fed could, in practice, attainthe benefits of makeup strategies that are possible in models.”I.5Cœuré (2013)“Most notably, the central bank may try to convince markets that it would keep interest rates
low, even if this would imply inflation well above its previous objective, at least temporarily.
The promise of higher future inflation, if credible, induces private agents to substitute future
for current consumption, hence providing additional stimulus today. This type of forward
guidance is closer to the academic concept of forward guidance in the strict sense—as discussed, for example, in Woodford (2012).
The main challenge of such guidance is its inherent inconsistency over time andthus lack of credibility. When the time comes, the central bank may be tempted
to deviate from its prior commitment: once the benefits of higher inflation expectations in terms of front-loaded spending have been reaped, the central bank
may not be willing to pay the bill in terms of higher inflation afterward. If the
public foresees this temptation, expectations might remain unaffected in the first
instance and the desired inter-temporal substitution of spending might not materialise. This is a possible explanation why, in practice, central banks have refrained fromusing forward guidance in a way that implies a major change in strategy. Therefore, central
banks’ forward guidance has rather aimed at providing greater clarity on the reaction function
and the assessment of future economic conditions.”

56

I.6

Dudley (2013)“With respect to forward guidance, it is important to distinguish between two specific forms
that this guidance may take. In the first form the central bank provides its forecast for the
future path of the policy rate and, possibly, some sense of the degree of uncertainty aroundthis path. In the second, the central bank pre-commits to a specific future path for its policy
rate.
Providing a forecast for the policy rate by itself does not create any budget or reputationalrisk for the Federal Reserve, so I generally do not see the first form of forward guidance as
posing much risk to central bank independence.
The second form of forward guidance—pre-commitment to a policy rate path—could
create more risk for the central bank. In particular, consider a scenario in which
the central bank decided to increase monetary accommodation by committing tomaintain a low short-term interest rate for a long time even if this commitment
resulted in inflation overshooting the central bank’s objective in the future. I
could see how this could create a potential threat to the central bank’s independence. That is because the commitment could force the central bank in the future
to conduct monetary policy in a way that was inconsistent with the inflation portion of its mandate.
Although this second form of forward guidance could create greater risk for the central bankwith respect to its future independence, this is not a policy that has been adopted by theFederal Reserve. There are implementation challenges with this approach. In particular,
it is difficult for a monetary policy committee today to institutionally bind future monetary policy committees to follow actions that could conflict with theirobjectives in the future. Without such a credible forward commitment, such policies would likely be ineffective in affecting expectations in the manner needed toprovide additional monetary policy accommodation.”

I.7

George (2019)“Third, a price-level targeting strategy is time inconsistent unless policymakers can crediblycommit to following it. If the goal is to have inflation of 2 percent on average, a period of
below 2 percent inflation would require an equal period of inflation above 2 percent. But
once inflation has moved up to 2 percent, policymakers might be tempted torenege on their prior commitment and not allow inflation to go higher. This
would undermine the future credibility of the price-level targeting strategy. To
the extent the public understood this time inconsistency problem, price-leveltargeting would not be credible to begin with, absent a commitment device.
With regular turnover among members of the FOMC, it would be difficult for
one Committee to commit a future Committee to a particular course of action.”

I.8

Lacker (2013)“Designing such conditional guidance involves trade-offs, however. Credibility requires consistency, over time, between a central bank’s statements and its actual subsequent actions. A
57

central bank’s statements will have greater immediate effect on the public’s expectations themore they are seen as limiting the central bank’s future choices. Yet there are likely to be circumstances, ex post, in which the central bank feels constrained by past statements. Yielding
to the temptation to implicitly renege by reworking decision criteria or citing unforeseen economic developments may have short-term appeal, but widely perceived discrepancies between
actual and foreshadowed behavior will inevitably erode the faith people place in future central
bank statements. So central banks face an ex ante trade-off, as well, between the short-run
value of exercising discretion and the ability to communicate effectively and credibly in the
future.”

I.9

Plosser (2013)“Note, however, that the central bank’s ability to influence the public’s belief about the future
path of policy and the economy depends critically on the bank’s commitment to that policy
path and the credibility of that commitment in the eyes of the public. The public mustbelieve that even after the economy begins to strengthen, the central bank willhold rates lower than it otherwise might have found desirable to do had it notbeen at the zero bound in the past."

I.10

Powell (2019)“By the time of the crisis, there was a well-established body of model-based research suggestingthat some kind of makeup policy could be beneficial. In light of this research, one might ask
why the Fed and other major central banks chose not to pursue such a policy. The answer liesin the uncertain distance between models and reality. For makeup strategies to achieve theirstabilizing benefits, households and businesses must be quite confident that the “makeup
stimulus” is really coming. This confidence is what prompts them to raise spending andinvestment in the midst of a downturn. In models, confidence in the policy is merely anassumption. In practice, when policymakers considered these policies in the wakeof the crisis, they had major questions about whether a central bank’s promiseof good times to come would have moved the hearts, minds, and pocketbooksof the public. Part of the problem is that when the time comes to deliver the
inflationary stimulus, that policy is likely to be unpopular–what is known as thetime consistency problem in economics.”

I.11

Ueda (2013)“If Max (the Taylor rule rate, zero) describes the usual central bank’s reaction function to themacroeconomic environment, the central bank can generate easing effects by offering a newreaction function to the market with a promise of a longer period at the zero rate than theabove rule suggests. To the extent that the Taylor rule represents an optimal response of the central bank to macroeconomic environment, however, this forward
guidance strategy amounts to “irresponsible” central bank behaviour. In otherwords, the strategy is time-inconsistent. This means that when the economy nolonger requires a zero rate, it is better to raise the interest rate, reneging on thepromise made. If people foresaw this ex ante, however, the strategy would become ineffective. Thus, the central bank would be sending a confusing signal if it was using
58

forward guidance in this sense and insisted that it was still behaving in a “responsible” way.
Also, the central bank does not seem to get much mileage out of a vague promise, such as themaintenance of a low policy rate “for an extended period,” unless there is much confusion in
the market as to where the policy rate would go in the short term.
The BOJ seems to have faced the time-inconsistency problem in 2000.”

I.12

Williams (2012)“Although forward policy guidance has proven to be a very useful policy tool, it’s not a perfect
substitute for the kind of monetary stimulus that comes from lower interest rates. One issue
is that, for the forward guidance policy to work as desired, the public has to believe that the
FOMC will really carry out the policy as it says it will. But, the Fed doesn’t have the ability
to tie its hands that way. This point was made by Finn Kydland and Edward Prescott in
the late 1970s. Let me explain. For forward policy guidance to have its maximum effect,
the Fed must commit to keeping the short-term policy rate lower than it otherwise would
to compensate for the fact that the short-term interest rate cannot be lowered today. But
when the time comes to carry out the commitment made in its forward guidance,
it may no longer want to do so. For instance, it might be hard to resist raisingrates earlier than promised to head off an increase in inflation. So, even whencentral bankers say they will keep rates unusually low for a set time, the public
may worry that the central bank will raise rates earlier to fight budding inflationpressures.”59